Multiple ways to multiply

So … dh commented to me that dd told him that she did not know the 6 times table. Worried, he told me that while it is good that she explores freely and deeply etc , we have to ensure that she doesn’t miss basic things.

I told him, you know the funny thing is, she was telling me something about 6 x 8, which she found difficult because she knew neither the 6 table nor the 8 table. She said, “Well, 6 x 8 is so hard that I just memorized it.”

This was somewhat surprising to me because I thought, what other way is there? Don’t we memorize all of them? Read the rest of this entry »

Never Enough Time

In my younger and more flippant days, I liked to call my chosen approach to homeschooling, "do nothing" after Fukuoka’s method of farming.

Japanese farmer Masanobu Fukuoka practices what he calls the "no-plowing, no-fertilizing, no-weeding, no-pesticides, do-nothing method of natural farming." To him it is ego-centric to think that people grow crops. Ultimately it is nature that grows crops.

– from Green University

Calvin: There's never enough time to do all the nothing you want.

Do you have enough time to do nothing?
from Bill Watterson, The Authoritative Calvin and Hobbes, p213

Slow Learning

We often ask, what is learning? Now let us ask, what is slow learning?

1. Slow

In Space and Time in Classical Mechanics, Einstein asks to imagine that he has dropped a stone while in a moving train.  As it happens he asks us to imagine that he has dropped it outside the train, from the window, as the train movedi.

Inside a moving train, if we drop a stone we will see it fall down in a straight, vertical line.  If we are inside the moving train but drop the stone outside the train, we will see the same thing.  To the falling stone, once released from Einstein’s (or anyone’s) hand, it makes no difference whether it is inside or outside the train.

An observer outside the train, on the platform, (or on the embankment, as in Einstein’s tale), will see the stone come down in a parabolic path.  As if it were not merely dropped but thrown.  To those inside the train, moving forward at the same rate as the stone itself moves forward, the forward motion of the stone is invisible.  We might say it is non-existent or cancelled out, like the motion of the earth – which we do not count we are sitting still.  Or when we drop a stone while sitting still.

Now the question Einstein asks us is:  What did the stone do? Did it fall in a straight line or along a curve?

As Einstein goes on to explain in the rest of the book, Relativity: The Special and General Theory, questions of speed, distance and time become relative to the frame of reference.

Learning also takes many paths, perhaps all paths, as the quantum physicists say of particles. Is one path longer than another? Faster?

What parent or teacher is not familiar with this experience – in a conversation with a child, a flurry of ifs and buts arise, so that a simple point that that you thought you would explain in five minutes gets deferred for hours or days. Meanwhile as you follow the tangents, further questions arise. Is your original question forgotten? No, it is still out there, drawing you towards it via this loopy, squiggly, elliptical path. Teaching that is based on a fixed notion of the direct path may not allow for such digression. It may even subtly discourage it – akin to the terse “recalculating” one hears from navigation instruments in the car when one has veered away from the designated route. Yet the curiosity of children will keep these questions alive, patiently or impatiently awaiting their turn on the front burner.

If it takes two days to communicate a point that you thought would take five minutes, do you feel that time has been lost?  What happens when teaching complex concepts and skills – what if your child learns something months or years after the expected date? Sometimes people who want to trust the journey of learning find themselves wondering,

Is this child slow?  Is s/he falling behind?  Will it be difficult to catch up later?  Will it hurt if I push her or him?  At what point should I intervene?

Many people have written about these questions with well-reasoned points and evidence supporting a spectrum of approaches. Some suggest creative ways to encourage progress, indicators for intervention when there is no progress, or reassurance that it will happen in its own time.

Some will say, “they will learn it when they learn it.” Some will say, “they will learn it when they need it.”

But what is it? Do we know?

Maybe we do. Or maybe we only think we know.

Is my understanding, Wittgenstein asked, blindness to my lack of understanding? Often, he continued, it seems so to meii.

Me too! How often in school had I felt that I was expected to understand or at least pretend to understand something, glossing over whatever had made me reluctant to accept it. Years later, when I read Wittgenstein this sentiment helped me to understand my frustration, and reclaim my doubts throughout discussions of such concepts as normal force in physics or set in mathematics, considered too straightforward to interrogate.

“A set is a collection of objects,” I recall a teacher explaining. “So here, see this picture of various objects. I will draw a circle around it, now it is a set.”

Simple! But I did not understand. What exactly made this a set? Thinking back, it would probably have helped if someone had said, “set is a complex idea, there is a lot more to learn about it and some of it doesn’t even make sense, but that is part of what makes it fascinating. Today we are going to look at some simple examples, where we just circle things, call them sets, and consider how some sets are similar or different from others.”

“What makes something different?” That would have been my next question. Even if we didn’t pause right away to go down the roads exploring the definitions of set, same, different, it would have helped if we acknowledged that these were uncertain, and depended on more factors than we could ever pin down, rather than pretending that we knew just what they were and that the sensible response to a worksheet on the topic of sets was to circle the objects as instructed.

Such an acknowledgment is not unheard of. In fact, I can never forget the preface to my sixth grade math book. Meant for the teacher, it stated that “In this book we do not prove the commutative property or the identity property.” Amazing! So even statements like a=a or a+0=a or a+b=b+a could be proved, meaning they could also be questioned. How happy I was that the authors of the math textbook had chosen to confide this information in me.

Had they not done so, and regarded these equations as obvious, requiring no proof, then any question about them would have been regarded as pure nonsense, unthinkable. Now that they had acknowledged that it was indeed thinkable, not only could I patiently wait for higher level classes where we would be encouraged to think about such questions, but I could also have faith that the math I was part of something deeper, that touched the heart of what it meant to say a=a, indeed, what meaning was.

2.  Learning

Let me tell a story about our daughter and the (recently glamourous) subject of arithmetic.

From as far as we can remember, our daughter delighted in number, shape, order, series and various mathematical concepts.   She would observe shapes and patterns and then one fine day tell us something about them that would wow us.  She was equally thrilled to hear about math.  Indeed she heard math in places we would not have expected, casually comparing a musical piece to a multiplication process.

Everything reminded her of math.  She knew it too, and delighted in it.  While arranging her clothes in her shelves she referred to priority and order of operations.  While overhearing us refer to combinations and permutations in the context of tracing old classmates she immediately corrected us – “you can’t have permutations!”  Seeing our blank looks, she explained, “what would they do, enter the room in a different order?”

When it came to basic sums, though, she added on her fingers most of the time.  Would this be considered late?  Slow?


The Addition Table

The Addition Table

One day she arranged her dominoes in a pattern and called me to see that it served as an addition tableiii.   She arranged the dominoes such that you find the two numbers that need adding in their respective row and column, find where they intersect, and then count up all the dots on that domino. Most of us who do one-digit addition without thinking about it would find this more time consuming. If she had learned addition by heart then would she have ever devised this addition table? Arranged in various patterns, the dominoes illustrate concepts that might take a greater understanding of math or number theory to describe in words. And they get to the heart of what it means to add.  (She has demonstrated here.)

I bring this example up because when we talk about how unschooling facilitates learning at one’s own pace, most people think it means that we need to be patient with “slow” learning but rarely we get an example of learning that is made possible precisely because something else was not yet learned or was learned “slowly.”

If we rush to “understand” addition, as indicated by correctly and promptly adding given numbers, we may miss out on investigating what addition is, and what numbers are.

Had she memorized her basic addition facts, would she have devised an addition table?  Perhaps.  When?  Would that have been considered late?  Or slow?

What did she learn by making the addition table?  How was this learning facilitated by the fact that addition had not yet been ticked off her list of skills to master?

Learn as if you would live forever, said Mahatma Gandhi.  Not only will you be unafraid to learn something new, you will be unafraid not to know, and unafraid to say “I don’t know.” You will not fake it, you will not be rushed to learn something when you are arrested by something more fundamental.  And as we approach the answer to one question we may again find our path slowed by still further questions.

For example – when coming across the phrase “first prime minister” (of India), my daughter was not interested in the name corresponding to this epithet.  She wanted to know what this phrase meant.  A question about what the “first” of a kind could be, how a given specimen could be “first” of a kind at all.

Her question:  So did they already decide to call the person a Prime Minister?

As I collected my thoughts to answer, there came another question – But who, they?

A question about the nature of authority itself, who vests it in whom.   (Is this history?  Or math?  Or politics?  Or philosophy?)

And then:  When did they call it India?

Those who “know” the answer to the question “Who was India’s first prime minister?” would probably answer the question, quiz-show style.

But how would they “know” such information?   And how would they “know” that one responds to a question with “the answer” rather than with further questions?

Slow learning empowers the learner over the learned and values the slow in the spirit of the movements for slow food, slow money and slow love.

Of slow love, it is said, “Slow love is about knowing what you’ve got before it’s goneiv.”

You can look up the name of the prime minister.  But when you stop asking questions about first-ness and prime-ness, where do you go to tap into your earlier wonder about these concepts?

i Albert Einstein, “Space and Time in Classical Mechanics” in Relativity: The Special and General Theory. 1920. Accessed online from on June 19, 2013.

ii Ludwig Wittgenstein, On Certainty, §418.

iii I have described this in a comment posted on Peter Gray’s article, “Kids Learn Math Easily When They Control Their Own Learning“ in his blog Freedom To Learn.

iv– Dominque Browning, Slow Love, pg. 5.

Slow Learning” also  appears on the website of Swashikshan: Indian Association of Homeschoolers.

Studying History

“Studying history is …. pretty much useless, isn’t it?” my daughter asked, as if it was fairly obvious.  “I mean why read about Columbus discovering America, when it isn’t even true?”

Alarmed.  All senses alert.
Mission: Rescue the field of history.   Why?  
Think.  Think.  
Meanwhile, don’t answer.   A weak response merely invites rebuttal.  Then what?  Stay calm.  Say nothing.  No one need get hurt.
Ah, now I remember what to do.   Listen. 
Do something.  Explain why we need to study history.   Justice – that’s right, we can start with justice.  Rosa Parks.  Freedom struggle.  Forest Rights.   Let’s just take care not get to too complicated.
The next morning as my daughter ate breakfast, I combed her hair.  I had my question ready and eased it ever so casually into the flow of braiding and talking. 
“Have you ever come across something that is not fair?”
“Yes,” she said, wailing.  She tends to take on the mood of whatever she is talking about.   “Like right now, I am finished with my oatmeal but you are only finished with one జడ (braid) and I don’t have anything to eat during the other జడ.”  
As I took that in, she asked, “why did you ask me that?”
“Oh,” I chirped cheerily, “I just wanted to hear what you thought.”  
“That is nice” she said, genuinely appreciating it.  Trust me, she is not one to say “that’s nice” for no reason. 
On that happy note, we carried on an easy banter as I finished up the other braid.  No further inquiry.
Throughout the day I kept my troubles regarding the beleaguered field of history to myself.  I think a couple of times she may have continued thinking aloud about how she did not like history.  It was almost as if she liked the sound of it:  “I don’t like history.”  I thought, but did not say, things like, “you like history, you are only saying that because you have gotten some idea that history is boring …. probably picked this up from some character in one of your books[1].”
Later that night she asked me, “what does Machiavellian mean?”  She was reading The Penderwicks on Gardam Street, in which Anna had used the word, saying only “I will explain later,” when Batty, another character in the novel, asked her what it meant.
I couldn’t resist.  I said, “well I can tell you but it might involve a little bit of … history.”  We both giggled. 
In small bits, I explained, proceeding further only as the questions directed. 
It reminded me of an incident some years ago when cleaning our pichkari in preparation for Holi.  Branded in the plastic were the words “Made in Mumbai.”   Mumbai … Bombay … English … British … the conversation went all over the world and back.  Language, culture, power, land, water, salt, spice, east, west … we are still talking about it.  
Last summer in Maryland, we saw a car with a Delaware license plate.  Across the top it said “THE FIRST STATE.”
“Why does that say Delaware – the first state?” my daughter asked me. 
By now it was a running joke:   “Just to let you know, this might involve a little bit of history.”  As it turned out, it involved politics and ontology as well.  
Were they already called states?  she asked.  Who named it Delaware?  Was it like New York – named by the British, before they decided not to be British? 
Or later when they decided to call it the United States did they realize that the first one was Delaware?    What did it do first to become the first state?   Is Delaware also a Native American name?  Did the Native Americans want to name the state?
One day we saw Bhakta Prahlad (1967).  While observing the crowning of Prahlad, she asked, “how did the first king become king?”
Oh, yeah …
*  *  *  *  *
Now you may ask, can you really pursue history through the questions that come up while walking around and doing other things?  What about the people to whom you ask the questions … would the meandering route get you anywhere if those people hadn’t ever learned about the European Renaissance, and how the American constitution was ratified?  At some point don’t you also have to do the heavy reading for yourself?
And till that fine day comes when your curiosity drives you to read those works, what can you do to prepare?  Sure, maybe you spend years randomly asking questions and getting quick fill-ins from Wikipedia and then suddenly start reading Romila Thapar or Amartya Sen or Howard Zinn or Victoria Brown.  I can well imagine doing that.  I could also imagine missing the boat entirely.  Do we leave it up to chance?  
Such questions come up from time to time, especially for those whose children are learning outside of school, and possibly without any standardized curriculum.  Is there a plan to the learning, or is it driven by whim and fancy?  Granted, whim and fancy can take you a long way, usually via the scenic route.  But what about structured learning?  I think that structure materializes in various ways, and if we observe we might find, for example, that a child’s mind goes into active mode while taking a bath or getting ready for bed.  Some kids think better when in motion, or when they are interrupting you (as opposed to when you ask them a question and wait for them to answer).  Some kids cry when they don’t understand something, but that crying is part of their grappling with the problem.  Recognizing and respecting the structure that goes into each child’s learning process will help to keep the channels clear and unobstructed. 
Apart from the patterns that emerge, are there practices that we can encourage, that will help to build our capacity to learn various things?
This brings me to another answer on structure.  Consider a subject like history.   History is not something that others create and we consume.   History must be questioned at every turn – who said? how do you know? what if…?   But how do we learn how to question history?  
In fact children do ask these questions about every day happenings.  We can draw upon this inquisitive nature and apply it to the study of the past as well.  At the very least we can avoid suppressing these questions when they arise.  Questioning authority, vital for investigating the past, starts with questions like “why do we have to do this?” that can spark intriguing discussions. 
I believe that one should learn history by learning to work like a historian, to think like a historian.   Before my child reads the history that is written by others, I would like her to try coming up with some of her own questions and finding something out about history.  We have, for example, interviewed people in the neighbourhood.  So far we have found out more about the mechanics of setting up and conducting an interview than about any particular historical or social question, but this is important too.  When we read interviews of others, it gives us some insight into what it takes to set the stage.  We have also looked around the house and neighbourhood and made a kind of game of asking, “what happened here in the last few hours or days and what is our evidence for saying so?”  “How reliable is this evidence?”  “What might possibly make us doubt this evidence?”  “What alternate explanations are there?”  
This practice will equip us to ask similar questions while reading history, so that we can evaluate sources and compare perspectives.  At a more elementary level, a game like Chinese whispers, illustrating how a message gets distorted as it passes through various transmitters (people) can also sharpen one’s critical faculties when reading others’ accounts of what happened. 
[1] Or was it in one of my books?  I had recently been reading James Loewen, Lies My Teacher Told Me.  Loewen writes,
“[P]rofessional historical organizations for at least a century have repeatedly exhorted teachers not to teach history as fact memorization. ‘Stir up the minds of the pupils,’ cried the American Historical Association in 1893; ‘avoid stressing dates, names, and specific events, ‘historians urged in 1934. . . . Nevertheless, teachers continue to present factoids for students to memorize.”  
I don’t think she read the book, but I am just saying, I need not turn to a juvenile novel to track down the sentiment that history class is boring.  Historians themselves have been complaining about it for years.

Right to Education

After reading Alfie Kohn What does it Mean to be Educated? and Jonathon Kozol On Being a Teacher I am all fired up. Must do something, etc etc.

A few weeks ago I was all fired up when my neighbour told me what happened to her son at school. The English Ma’am wrote something on the board. It contained an error, which the boy noted (aloud). The Ma’am reprimanded him and he remained silent.

Considering all the bad grammar I hear routinely, I guess I should not be too surprised at this. But an English teacher? This boy is an avid reader so I am sure he has an ear for correct English, certainly at the grade school level, and apparently more so than that of his teacher. The word in question was the plural of sheep which the boy correctly pointed out, was sheep. Now in this situation, did the teacher

a – acknowledge her error and appreciate the alert student who cared enough to point it out?
b – say, oh, is that so, let me look it up and confirm?
c – say, “Do you know better than your teacher?”

The correct answer: c. No prizes for guessing and no extra credit for realizing that the tone and volume in which it was asked meant that there was only one acceptable answer to that question. The boy said, “No, Ma’am” but later confided to his mother that he had wanted to say, “Yes, Ma’am.”

Is this not a violation of this boy’s right to education? He may be in school, attending class, doing homework, getting marks. But is his right to education respected? When he speaks is he heard with interest, with patience?  When he asks a question, is he encouraged? Answered? Or told not to ask? Even when he responds correctly he risks being silenced. Is honesty something to be read in the Morals class textbook1 or is one encouraged to speak the truth, even when your voice shakes2? What is the role of the teacher? What power did she hold that caused this boy to back down, even though he believed what he said? Grades? Punishment? I doubt he was thinking so far ahead. Being snarked at by the teacher when he was merely making a grammar correction, as he would expect her to do if he made an error, was no doubt disturbing enough that his only aim at that moment would have been to make it stop.

And what about something more consequential than the correct plural of sheep? What about questions of history or science or social studies where there can indeed be multiple approaches and viewpoints, leave alone errors in the textbook or in the teacher’s lecture? We have even faced ambiguity in the grade 1 math textbook! Do students stand a chance at having their voices heard on such matters?  Are students allowed to try out ideas or methods that they may eventually discard, i.e, make mistakes?  And what would we in fact prefer, that they think and develop a viewpoint, compare with others, examine consequences, test against evidence, revise and refine …. or that they repeat the views provided by a textbook author or a teacher? Doing the latter ostensibly requires less effort and yet most children, given the chance, will do the former unless constrained to do the latter, for the sake of the grade and the teacher’s approval. Such constraint, which is the norm in the vast majority of the schools in India, public, private, elite or international, is a stark violation of children’s right to education.

How do we want to read our history lessons? Ahimsa, satyagraha, are these formulae to be memorized for an exam? Or are they living principles?  Can we too speak truth to power? Can we unearth the lies told in our history books, as James Loewen has done for US History? Can we question authority? Are we willing to risk our comforts for our values? Or vice-versa?

Emphasis on rote learning and standardized testing widens the gap between the haves and have-nots, and not only because it renders students passive consumers of pre-fabricated knowledge.  The haves, after all have opportunities outside of school to explore ideas freely. They may converse daily with family and friends and at some point, usually quite early, recognize that school tests are a game they play and but a thin slice of the wide arc of learning that beckons.  Even if they have little time outside of school,  whatever original observations or experiments they make are duly recognized and valued.  In our neighbourhood the children have buried various objects, planted seeds, made up imaginary games and plays – when children in poorer communities do such things they may get reprimanded for “wasting time.”   In our neighbourhood when parents  come to drag their kids home from the playground, they do so wistfully, and often try to compensate for it during other times.  Even this modicum of awareness of the value of free play is denied to many children.

Standards are not confined to “the basics” of reading, writing and arithmetic but extend to social and economic values, which are increasingly dictated by corporate interests, either through direct intervention in school programming, or through emphasis on ranking and testing. Alfie Kohn talks about this: .

In India, where test-based teaching is unlikely to be dethroned anytime soon, even corporations are making fashionable noises against “rote learning.” They are not against standardized tests, mind you, they just believe they can get better results on those tests through hip means such as independent, innovative and critical thinking.

from The Unschool Bus

I expect you all to be independent, innovative, critical thinkers ….

Thanks to The Unschool Bus  for this apt illustration.

The press drums along with them without questioning the goals of education, appropriately shocked (shocked!) that children can’t keep their Gandhis straight or give liberal minded answers to questions on rights of women and immigrants. See The Hindu, 12 Dec 2011, “Learning by rote prevalent in top schools too.”  In the entire article neither the top-ness of these schools or the validity of the tests are called into question.  Nor are we invited to wonder why the children have “wrong” answers.  For the record I would be open to hearing about how the “shape of a square object would change if it is tilted,” as nearly half the sampled children apparently opined.

If the powers that be in the classroom are not answerable to the students, if the questioning goes only one way, and answers are determined by the questioner, then it is inevitable that the those who succeed in this path would need to keep it that way.

But do students, parents or teachers really believe that it should be that way? I don’t think so. We also believe that education and literacy enable us to go to the source, to look for evidence, to take it apart and see how it works and how it doesn’t, and even make our own models and theories and stories.  Insofar as everyone gains these abilities, no one can cheat another, and therefore we can, together, build a more just society.

Serious change is required for education and literacy to achieve their potential to act as tools of empowerment. Today we see them promoting conformity and consumerism and widening inequality. Right to Education, however, must not only mean right to be admitted in a school and consume the information and ideas dictated, and speak only on command, but right to express ideas, ask questions and actually learn without fear. On not need to be in school to exercise these rights, but certainly no school should infringe upon them. Outside of school, children do ask questions, question the answers, and even question the questions.  Right to education must include the right to make mistakes, not only the right to be right.   Unfortunately in school, the vast majority of students get little time or space to ask or even answer questions in their own words, or ponder questions tangential to what the teacher or test-book has asked.  Their role is to reproduce the answers provided to them. Children who step outside of this expected role will typically be punished by a bad mark, humiliation or even physical punishment. There is no grievance redressal mechanism for children whose right to education is violated in this way.

to be continued…

[1] In the closing scene of Bangaru Papa, Sekhar, regretting his inability to stand by his principles when his family prohibits him from marrying Papa on grounds of her caste and class, says his high status demands that moral values remain confined to the textbook.  See from 6:34 in this .
[2] Gray Panthers founder Maggie Kuhn is noted for saying, “Stand before the people you fear and speak your mind – even if your voice shakes.”

Listening and Learning from Birth

Listening and Learning from Birth

Foundations of natural learning begin at birth or even before. I am not talking about making the womb a classroom and reciting lessons for the benefit of the baby. Just as we feel that baby listens to everything going on, when we listen to baby we develop communication skills- both ours and baby’s. Many parents talk to baby during pregnancy. Mothers speak of tuning in closely to baby during labour and birth.

After birth, everyone tunes in. Long lost songs fill the air. What about these calls and responses from early days and years has to do with learning?

When we listen to children we model listening. It sounds obvious yet so often we see parents “telling” children to listen but not themselves listening. Children learn to listen by being listened to.

Listening to Infants

Adults can eat, sleep, and go to the bathroom without ever speaking to anyone. Not so for infants who need someone to feed them, hold them, breathe near them. And change them or better yet take them to relieve themselves. While most of the world still practices diaper-free hygiene and learns to listen to babies signal for this need just like all other needs, the rapid mass conversion to diapers is leading many to forget that babies actually do communicate hygiene needs from infancy. Fortunately many people are reviving this lost language and practicing what in the first world is known as “Elimination Communication.”

Elimination Communication

Compare a diaper to a curriculum: the lighter it is the more scope for the child to communicate. Disposable diapers or heavy plastic-covered cloth diapers, promoted for “not leaking” stand as a barrier to communication, like a rigid curriculum. To consider a diaper successful if it passes the “leak-proof” test is like a child considered successful based on passing an exam. And the more “leak-proof” is the diaper one uses, the longer it takes for the child to learn to go to potty on his or her own. In fact, babies are born with this awareness; constant use of the diaper makes them tune out and have to unlearn diaper-use and “learn” or “train” to recognize their own elimination needs again later.

Babies who are listened to, with respect to hygiene or any other aspect of living, will develop trust and communication skills more readily.

Just as diaper free hygiene helps a child stay in tune with her or his body, so many other aspects of natural family living such as natural birth, breastfeeding, unrushed transition to solid foods, sleep-sharing and respect for the body’s immune system help to draw an arc of holistic learning.

Precious Preliterate Years

When school started at 5 or 6 years of age and playschool was just that, then one left the early years free for learning that was more or less self-paced and self-directed. As it was natural to learn to walk and talk without instruction or monitoring, children had the opportunity to be exposed to letter, number, color, shape and other “pre-school” concepts as they occur in the surroundings without subjecting them to lessons and reviews and gold stars. Mixed age groups are more conducive for exploring social skills such as helping and taking turns that pre-schools claim to develop. Moreover, it is in these precious pre-literate, pre-numerate years that we experience the world as a whole, not squeezed into words, divided into parts and counted, compared, compiled. Why cut them short? Why not leave these years free for children to learn vital skills like eating and sleeping, at their own pace and with fewer demands on their time?

Leaving the early childhood years free of school schedules not only frees their minds from the impact of an external curriculum, but frees their time to learn life skills that may otherwise be shortchanged.

Freedom from Preconceived Notions

As deserving of our respect is the freedom the untutored mind has to explore and create knowledge as if they are the first to do so – as if our language, numbers, color schemes did not even exist.

Why do we make efforts to teach little kids to identify red, yellow and blue? Primary for whom? for what? In Telugu it is not easy to distinguish pink, orange and red, because all are called red. In fact we can’t even say “red” but have to say “redlike.” There are on the other hand very specific colours named “eggplant,” “peacock’s neck” or “parrot” and heaven help you if you call these violet or blue or green (esp if you are in the matching centre!) Some years ago I participated in an experiment for a study on “categorization of color*” and joined a listserv called colorcat that was dedicated to this research.

So even if we know that by age 5 or 6 or 7 children will have to accept the colours in standard crayon boxes why not leave the years before that free for them to perceive light and color in various ways without us dictating names for these things. I say the same for letters and numbers. Treasure those precious pre-literate and pre-numerical years without rushing to count, classify, compile and categorize. Though the languages* of infancy and early childhood may not survive for long, the opportunity to discover and develop and discard concepts in those years, the sense of being a principal investigator rather than passive recipient of concepts others have tried and found true, can serve one throughout life. Let us stand on our feet before we stand on the shoulders of giants. (*Wade Davis on endangered cultures gives us a glimpse of what values might survive along with non-dominant languages and concepts.)

Our daughter shared some memorable observations around age 2 about the number 0 and about white as a color. Yes, she currently uses our decimal number system and no longer objects to seeing colors the way we do. But we also got a glimpse of the questions our prevailing system raised for her and thought momentarily about what it would be like to see it her way. Alas, that moment may no longer be with us but the chance she had to protest the way zero was valued or the role of the white crayon is something that kindled in us respect for her imagination and analysis