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.

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.”