Homeschool+ Conference

Zero is Beautiful Presentation at the Homeschool+ Conference

Zero is Beautiful: Teaching Mathematics as if People Mattered

I presented my paper “Zero is Beautiful” at the recent Homeschool+ Conference, which is part of a series of conferences under the umbrella called “The Learning Revolution.”

It was really wonderful to have Dr. Maria Droujkova attend the session.

Here are the recordings of all the sessions including mine.  Read the rest of this entry »

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 »

Zero is Beautiful: Teaching Mathematics as if People Mattered

Forthcoming in Home Education Magazine, November-December 2014


 

Can you imagine the time before the discovery of zero? My husband and I got a glimpse of this when we witnessed the discovery of zero, not on the world-historical scale, but by our two-year-old daughter.

It was not an easy road. Counting had come uneventfully, but when numbers became numerals and the number 10 appeared on the page not with its own symbol, but with a 1 and 0, suddenly everything had changed. Till that moment, in her world it was still possible to have a system of enumeration like the one used by Ireneo Funes in Borges’ story, “Funes the Memorius.” Funes gives every number its own unique name. He has “an infinite vocabulary for the natural series of numbers” and no use for the concept of place value.

When our daughter saw that the numeral 10 comprised a 1 and a 0 she flung herself upon a chair and cried. Read the rest of this entry »

One real teacher

Excerpts from the essays of Paul Lockhart
Paul Lockhart teaches mathematics at Saint Ann’s School in Brooklyn, New York.


MC Escher, 1948

MC Escher, 1948

From letter to Keith Devlin, Mathematical Association of America

On teaching mathematics to young children:
“I want them to understand that there is a playground in their minds and that that is where mathematics happens.”


From Lockhart’s Lament, Mathematical Association of America

Mathematics is an Art

… if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the latter category.

Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. Read the rest of this entry »

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 http://www.bartleby.com/173/3.html 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.

What is number? and other Insights

When one appreciates the value of slow learning, one does not “teach numbers” or “teach letters” but simply notices the moments when one’s child stumbles upon these concepts and tries them out. Most likely one might miss “the moment” when a child first encounters these, if there is such a thing as the first encounter, but whenever it comes to one’s notice that the child is versed in these concepts, one can discover new things by observing how they approach and resist them, how they question and use them. Whenever this happens with my daughter, I tend to stop whatever I am doing and almost slow down the moment, just to make sure I don’t speed it up. I respond very slowly, refraining from giving any additional information, but asking questions in language as close as possible to what my daughter has used.

For example recently she said, “Do you ever wonder why you are you and not just someone playing with a doll that is you?” Whoa! I stopped in my tracks, as if into my lap had floated a gift wrapped in layers and layers of delicate paper … that one must open with care, so as not to disturb the next layer or remove too many layers at once, and risk missing some nuances of meaning.

Similar treasures are packed into questions and comments or even gestures, if we know how to notice them, of infants and toddlers.

– lines / mountains
– place value – value of the number zero.
– two, two …

Among the most precious gifts of slow learning is the chance to ask questions such as “what is number” and observe how someone who is pre-numerate approaches this concept. While today one might worry if their child “learned numbers” later than the expected average age, one may instead appreciate this rare opportunity to become privy to insights on such basic questions as “what is number?”

Elizabeth Spelke, psychologist, asks such questions:

‘What is number, space, agency, and how does knowledge in each category develop from its minimal state?’ ”

Here is the article.

New York Times

Profiles in Science | Elizabeth S. Spelke

Insights From the Youngest Minds

Erik Jacobs for The New York Times

Elizabeth S. Spelke: A video interview with the Harvard cognitive psychologist on babies and the nature of human knowledge.

By NATALIE ANGIER
Published: April 30, 2012 199 Comments

CAMBRIDGE, Mass. — Seated in a cheerfully cramped monitoring room at the Harvard University Laboratory for Developmental Studies, Elizabeth S. Spelke, a professor of psychology and a pre-eminent researcher of the basic ingredient list from which all human knowledge is constructed, looked on expectantly as her students prepared a boisterous 8-month-old girl with dark curly hair for the onerous task of watching cartoons.

The video clips featured simple Keith Haring-type characters jumping, sliding and dancing from one group to another. The researchers’ objective, as with half a dozen similar projects under way in the lab, was to explore what infants understand about social groups and social expectations.

Yet even before the recording began, the 15-pound research subject made plain the scope of her social brain. She tracked conversations, stared at newcomers and burned off adult corneas with the brilliance of her smile. Dr. Spelke, who first came to prominence by delineating how infants learn about objects, numbers, the lay of the land, shook her head in self-mocking astonishment.

“Why did it take me 30 years to start studying this?” she said. “All this time I’ve been giving infants objects to hold, or spinning them around in a room to see how they navigate, when what they really wanted to do was engage with other people!”

Dr. Spelke, 62, is tall and slim, and parts her long hair down the middle, like a college student. She dresses casually, in a corduroy jumper or a cardigan and slacks, and when she talks, she pitches forward and plants forearms on thighs, hands clasped, seeming both deeply engaged and ready to bolt. The lab she founded with her colleague Susan Carey is strewed with toys and festooned with children’s T-shirts, but the Elmo atmospherics belie both the lab’s seriousness of purpose and Dr. Spelke’s towering reputation among her peers in cognitive psychology.

“When people ask Liz, ‘What do you do?’ she tells them, ‘I study babies,’ ” said Steven Pinker, a fellow Harvard professor and the author of “The Better Angels of Our Nature,” among other books. “That’s endearingly self-deprecating, but she sells herself short.”

What Dr. Spelke is really doing, he said, is what Descartes, Kant and Locke tried to do. “She is trying to identify the bedrock categories of human knowledge. She is asking, ‘What is number, space, agency, and how does knowledge in each category develop from its minimal state?’ ”

Dr. Spelke studies babies not because they’re cute but because they’re root. “I’ve always been fascinated by questions about human cognition and the organization of the human mind,” she said, “and why we’re good at some tasks and bad at others.”

But the adult mind is far too complicated, Dr. Spelke said, “too stuffed full of facts” to make sense of it. In her view, the best way to determine what, if anything, humans are born knowing, is to go straight to the source, and consult the recently born.

Decoding Infants’ Gaze

Dr. Spelke is a pioneer in the use of the infant gaze as a key to the infant mind — that is, identifying the inherent expectations of babies as young as a week or two by measuring how long they stare at a scene in which those presumptions are upended or unmet. “More than any scientist I know, Liz combines theoretical acumen with experimental genius,” Dr. Carey said. Nancy Kanwisher, a neuroscientist at M.I.T., put it this way: “Liz developed the infant gaze idea into a powerful experimental paradigm that radically changed our view of infant cognition.”

Here, according to the Spelke lab, are some of the things that babies know, generally before the age of 1:

They know what an object is: a discrete physical unit in which all sides move roughly as one, and with some independence from other objects.

“If I reach for a corner of a book and grasp it, I expect the rest of the book to come with me, but not a chunk of the table,” said Phil Kellman, Dr. Spelke’s first graduate student, now at the University of California, Los Angeles.

A baby has the same expectation. If you show the baby a trick sequence in which a rod that appears to be solid moves back and forth behind another object, the baby will gape in astonishment when that object is removed and the rod turns out to be two fragments.

“The visual system comes equipped to partition a scene into functional units we need to know about for survival,” Dr. Kellman said. Wondering whether your bag of four oranges puts you over the limit for the supermarket express lane? A baby would say, “You pick up the bag, the parts hang together, that makes it one item, so please get in line.”

Babies know, too, that objects can’t go through solid boundaries or occupy the same position as other objects, and that objects generally travel through space in a continuous trajectory. If you claimed to have invented a transporter device like the one in “Star Trek,” a baby would scoff.

Babies are born accountants. They can estimate quantities and distinguish between more and less. Show infants arrays of, say, 4 or 12 dots and they will match each number to an accompanying sound, looking longer at the 4 dots when they hear 4 sounds than when they hear 12 sounds, even if each of the 4 sounds is played comparatively longer. Babies also can perform a kind of addition and subtraction, anticipating the relative abundance of groups of dots that are being pushed together or pulled apart, and looking longer when the wrong number of dots appears.

Babies are born Euclideans. Infants and toddlers use geometric clues to orient themselves in three-dimensional space, navigate through rooms and locate hidden treasures. Is the room square or rectangular? Did the nice cardigan lady put the Slinky in a corner whose left wall is long or short?

At the same time, the Spelke lab discovered, young children are quite bad at using landmarks or décor to find their way. Not until age 5 or 6 do they begin augmenting search strategies with cues like “She hid my toy in a corner whose left wall is red rather than white.”

“That was a deep surprise to me,” Dr. Spelke said. “My intuition was, a little kid would never make the mistake of ignoring information like the color of a wall.” Nowadays, she continued, “I don’t place much faith in my intuitions, except as a starting place for designing experiments.”

These core mental modules — object representation, approximate number sense and geometric navigation — are all ancient systems shared at least in part with other animals; for example, rats also navigate through a maze by way of shape but not color. The modules amount to baby’s first crib sheet to the physical world.

“The job of the baby,” Dr. Spelke said, “is to learn.”

Role of Language

More recently, she and her colleagues have begun identifying some of the baseline settings of infant social intelligence. Katherine D. Kinzler, now of the University of Chicago, and Kristin Shutts, now at the University of Wisconsin, have found that infants just a few weeks old show a clear liking for people who use speech patterns the babies have already been exposed to, and that includes the regional accents, twangs, and R’s or lack thereof. A baby from Boston not only gazes longer at somebody speaking English than at somebody speaking French; the baby gazes longest at a person who sounds like Click and Clack of the radio show “Car Talk.”

In guiding early social leanings, accent trumps race. A white American baby would rather accept food from a black English-speaking adult than from a white Parisian, and a 5-year-old would rather befriend a child of another race who sounds like a local than one of the same race who has a foreign accent.

Other researchers in the Spelke lab are studying whether babies expect behavioral conformity among members of a group (hey, the blue character is supposed to be jumping like the rest of the blues, not sliding like the yellow characters); whether they expect other people to behave sensibly (if you’re going to reach for a toy, will you please do it efficiently rather than let your hand meander all over the place?); and how babies decide whether a novel object has “agency” (is this small, fuzzy blob active or inert?).

Dr. Spelke is also seeking to understand how the core domains of the human mind interact to yield our uniquely restless and creative intelligence — able to master calculus, probe the cosmos and play a Bach toccata as no bonobo or New Caledonian crow can. Even though “our core systems are fundamental yet limited,” as she put it, “we manage to get beyond them.”

Dr. Spelke has proposed that human language is the secret ingredient, the cognitive catalyst that allows our numeric, architectonic and social modules to join forces, swap ideas and take us to far horizons. “What’s special about language is its productive combinatorial power,” she said. “We can use it to combine anything with anything.”

She points out that children start integrating what they know about the shape of the environment, their navigational sense, with what they know about its landmarks — object recognition — at just the age when they begin to master spatial language and words like “left” and “right.” Yet, she acknowledges, her ideas about language as the central consolidator of human intelligence remain unproved and contentious.

Whatever their aim, the studies in her lab are difficult, each requiring scores of parentally volunteered participants. Babies don’t follow instructions and often “fuss out” mid-test, taking their data points with them.

Yet Dr. Spelke herself never fusses out or turns rote. She prowls the lab from a knee-high perspective, fretting the details of an experiment like Steve Jobs worrying over iPhone pixel density. “Is this car seat angled a little too far back?” she asked her students, poking the little velveteen chair every which way. “I’m concerned that a baby will have to strain too much to see the screen and decide it’s not worth the trouble.”

Should a student or colleague disagree with her, Dr. Spelke skips the defensive bristling, perhaps in part because she is serenely self-confident about her intellectual powers. “It was all easy for me,” she said of her early school years. “I don’t think I had to work hard until I got to college, or even graduate school.”

So, Radcliffe Phi Beta Kappa, ho hum. “My mother is absolutely brilliant, not just in science, but in everything,” said her daughter, Bridget, a medical student. “There’s a joke in my family that my mother and brother are the geniuses, and Dad and I are the grunts.” (“I hate this joke,” Dr. Spelke commented by e-mail, “and utterly reject this distinction!”)

Above all, Dr. Spelke relishes a good debate. “She welcomes people disagreeing with her,” said her husband, Elliott M. Blass, an emeritus professor of psychology at the University of Massachusetts. “She says it’s not about being right, it’s about getting it right.”

When Lawrence H. Summers, then president of Harvard, notoriously suggested in 2005 that the shortage of women in the physical sciences might be partly due to possible innate shortcomings in math, Dr. Spelke zestily entered the fray. She combed through results from her lab and elsewhere on basic number skills, seeking evidence of early differences between girls and boys. She found none.

“My position is that the null hypothesis is correct,” she said. “There is no cognitive difference and nothing to say about it.”

Dr. Spelke laid out her case in an acclaimed debate with her old friend Dr. Pinker, who defended the Summers camp.

“I have enormous respect for Steve, and I think he’s great,” Dr. Spelke said. “But when he argues that it makes sense that so many women are going into biology and medicine because those are the ‘helping’ professions, well, I remember when being a doctor was considered far too full of blood and gore for women and their uncontrollable emotions to handle.”

Raising Her Babies

For her part, Dr. Spelke has passionately combined science and motherhood. Her mother studied piano at Juilliard but gave it up when Elizabeth was born. “I felt terribly guilty about that,” Dr. Spelke said. “I never wanted my children to go through the same thing.”

When her children were young, Dr. Spelke often took them to the lab or held meetings at home. The whole family traveled together — France, Spain, Sweden, Egypt, Turkey — never reserving lodgings but finding accommodations as they could. (The best, Dr. Blass said, was a casbah in the Moroccan desert.)

Scaling the academic ranks, Dr. Spelke still found time to supplement her children’s public school education with a home-schooled version of the rigorous French curriculum. She baked their birthday cakes from scratch, staged elaborate treasure hunts and spent many days each year creating their Halloween costumes: Bridget as a cave girl or her favorite ballet bird; her younger brother, Joey, as a drawbridge.

“Growing up in my house was a constant adventure,” Bridget said. “As a new mother myself,” she added, “I don’t know how my mom did it.”

Is Dr. Spelke the master of every domain? It’s enough to make the average mother fuss out.

NEW EXPERIENCES Elizabeth Spelke with her daughter, Bridget, in France in 1988, on one of many family trips.

Enlarge This Image

DAUGHTER Bridget Spelke, who is a medical student, in South Africa. “My mother is absolutely brilliant, not just in science, but in everything,” she said.

Readers’ Comments

A version of this article appeared in print on May 1, 2012, on page D1 of the New York edition with the headline: From the Minds of Babes.

Profiles in Science

Elizabeth S. Spelke

This is the ninth in an occasional series of articles and videos about leaders in science.

Home

Which Bird is Closer to Home?

Which Bird is Nearer to Home?

Children, please open your Math-Magic Books and turn to page 6.

Now which bird is nearer to home? 

Child points to brown bird.

Why?

Shows with finger the path from the bird to the tree.    Both bird and tree are slightly in the background.  The bird has a slightly lost expression, but seems to be headed towards the tree.

Showing the lavender bird, “What about this one?”

Child points to the brick house and says, “that is not the bird’s home.”   This bird is headed straight for the neatly thatched roof of the brick house.

* * * * *

Above is what actually happened one day, years ago, when I brought out the Math-magic book and looked it over with my daughter.  I could have gone on to show that even if we were talking trees, the lavender bird is also closer to the trees.   But that would have just been contrived, as if I was trying to get the answer expected by the book.   And it is not even entirely clear that the lavender bird is closer to the tree in the background – even if on paper the linear distance is less, the perspective in the drawing suggests that one tree is further from the viewer than the other.  Same with the birds.  Which tree is home to which bird?  We don’t know.  The drawing does not give us this information.  In retrospect, it also seems likely that in judging “close to home” she looked not only at the scalar distance, but also at the direction in which the bird was travelling.   It may be near the tree, but if it is flying towards the mud house, then it is closer to home than another bird which is actually flying towards the tree?

What is clear (to the experienced worksheet user) is that the textbook intends for us to recognize the thatched-roof house as the “home” and is asking which bird is closer to that structure.  The “right” bird is aiming for that roof with a bright expression.  The “wrong” bird” looks lost and far away.  At least these are the clues that must have indicated to me which bird we were expected to circle.  I did not take into account the depth in the drawing nor ask where the bird made its home.  Only because our answers differed did I examine my thoughts at all – had she circled the lavender bird, I suppose we would have forged ahead to Tick (✓) the Cat Farther from the Tree.

Since I asked why, rather than moving to “correct” her, I learned something about perspective – in drawing and in our relationship with nature.

Tick the Cat Farther from the Tree

Tick the Cat Farther from the Tree

Images from Math-Magic Class 1 NCERT Textbook, page 6. Accessed online.

What is learning?

What is learning?

A child says, while pointing to objects consecutively, "two, two, two"

Observing this, her parents point out a picture of an octopus in a book and ask her how many arms it has. She obliges, "two, two, two, two, two, two, two, two!" The last "two" is said with a flourish, as if to report the total – or that is how it sounds to us, since we are used to counting in this cadence.

This is precious baby talk and much as her parents might delight in it, she will (all too quickly) grow out of it. The child, in this case, is the daughter of my friends, and she recently turned two. It seems she has been indicating things and saying "two-three, two-three."

The day will come when she abandons her system and counts as we do – 1, 2, 3, 4, 5!

Now what is learning …

When she said "2, 2, 2, 2, 2!" was that learning?

Or when she started saying 2-3 -2-3 was that learning?

Or when she says 1, 2, 3, 4, 5 will that be learning?

And what was she learning, and when? The concept of number? Of quantity?

I think that what she learned first of all was the concept of difference. Of non-one-ness. So in fact the word "two" did not signify the quantity two, so much as it signified "another." Now hear her "count" or point out the arms of the octopus: "Another, another, another, another, another, another, another, another!"

Another, again, more .. such words work magic for the newly verbal. They can serve to name anything, provided one of said thing is there to serve as a reference.

And when finally one replaces "another" with respective names, and two with three and four … what is one learning then? The names of the things, in the prevailing language. In order that others may understand what one has already understood, but has to translate into their language.

Read more: Two, another two and now there is a three