A parent's story.


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I became a surrogate mother streaming movies instead of milk.

Once I made a little girl stack up wooden rings upside down, every day, for 120 days in a row.

Put down the phone please. Don't call anybody yet. Let me explain.

The mother of this little girl did not speak to her much. The girl listened to TVs, computers and countless talking toys. She had a big brother and a big sister, but they had no time and patience to chat with a toddler.

By the age of two, the little girl - let me call her TLG - found that her body could produce some likeness of English speech. She was only struggling to understand how the big people were using those sounds.

From my previous observations of her siblings, I knew what was going to happen. As soon as TLG quit nursing and mastered the potty, her mom would put her in a preschool, where she would quickly start speaking English and become a good student. She would never try to learn her mom's native Russian, but she would never need it anyway.

There was at least one speechless year between now and then, and I did not like it. I did not like it to happen to her siblings either, but I did not know what to do.

First of all, it was not my turn. In my bygone culture, children younger than three years belonged to their mom. Our moms had one overwhelmingly effective tool. Usually two. You know what I mean. As long as they wielded them, no man could match their natural superpower.

I could and did everything a mom could do, except this. Consequently, I knew much more about two year olds' bottoms, than about their tops.

Another reason holding me back was my age. I had learned many things and taught them many people, big and small, but I assumed that in the 21st century, I was no longer fit to teach the young citizens of a country as advanced as the US.

In this little girl's eyes, I was utterly useless, until she figured that I could get her a movie. Whenever her mom was busy or not in the mood to serve her non-essential needs, she climbed to my room holding the disk she wanted to watch.

I am not a fan of free sweets and entertainment, especially when it comes to little girls, and I saw an opportunity in her visits. She wanted a movie - let her run for it.

I understand that this true story may seem strange at best, but could you please take it as a geometry problem? Here is what I had, there is what I wanted. Don't ask what size or color was the triangle, for I won't tell. Just read on.

In the local warehouse club I picked up an "edutainment" series of DVDs, a typical mix of learning and fun. I had no expectation for the first. I knew that American kids take the fun and leave the learning to rot. I just hoped that TLG's mom could not possibly object an extra dose of this computer-generated Mickey Mouse and his gang.

Online selection of toys was huge and veeery boring. The more developmental they claimed to be, the less attractive they were. I ended up wondering if anybody was selling the immortal toddlers' classic... what did they call it in this country?

It turned out, the name of the toy was stacking rings, or just stackers. I found some, looked around and suspected I was taking a wrong turn. The toy was unpopular, if not to say disregarded. Well, after all, the best toy for a child has always been the parent.

At that moment I did something uncommon for the first time. I ordered two sets: one for TLG and one for me. Not because I liked them - I did not, I just chose the lesser of the available evils - but because I had a plan.

I wanted TLG to watch and follow me in doing something until we got it done. I had found this to be a problem with her siblings long before she was born. Not sure how common it is. I suspect it's very common because American teachers routinely evaluate the ability of their students to follow instructions. My American kids were resilient to my attempts to teach them manual skills to a degree I'd never seen before.

The next time TLG came to me with a movie, I said I wanted to play with her first, and unpacked my purchases. She was captivated for a moment. I spilled off the rings, took one of them and said in my best English: "Please find a big red ring like this and put it on the stick". So the story took off.

TLG quickly understood why people speak and started doing it on purpose. Soon I heard her yelling: "I will never play math with you! Ever!". I was impressed with her English. I did not know she learned it from SpongeBob.

I was ears-deep into the teaching game by then, contriving new challenges (I would have never thought how many opportunities are hidden between the stacking rings), spotting and exploiting her weaknesses, looking for rewards. A child is the best toy for a parent, after all.


We discovered the game of challenges, and then the pegboard.

This is a "bridge" chapter. I wrote it specifically for this demo, with the sole purpose of connecting the dots.

Playing through the stacking rings took us four months because I kept doing uncommon things. Normally, we parents drop the toys on the kids to turn our backs to them, not to mention that stacking rings just got to be assembled like shown on the box.

Instead, I was spending half an hour a day challenging TLG to stack up the rings in increasingly hard-to-copy ways. The picture shows a reenactment. I did not document our lessons until much later, and then I asked TLG to show her stacking skills too. She made a mistake but found it on her own.

It was one of the most interesting jobs in my life. I learned more about the developing mind than I had ever known.

Before ditching the rings, I spent some time playing around colors. Turned out, TLG knew them already. Before my first American kid went to American preschool, I thought this was normal, but then, seeing how much resources and attention are dedicated to teaching colors in this country, I suspected that Americans are born colorblind.

TLG certainly was not, and speaking English was not a problem anymore, but I clearly saw that working on my challenges was shaping her character. Before she turned three, she was a fighter and a problem solver. I wanted to continue the lessons, and I was looking for another toy.

TLG was the third child. The whole household was a box of toys, but none of them was what I wanted. I added and tried several new sets, finding them either not challenging enough or too challenging, and often challenging not in the right way. I was looking for a source of challenges which would be rich, simple, safe and easy to operate.

Two more months passed, and, out of desperation, I ordered two sets of a toy, which I did not like. In the pictures it looked ugly and stupid. It was in production for a long time, and every American parent must have been familiar with it. I just was not American parent.

The toy's official name is Lauri Tall-Stacker Pegs Building Set. The box contains a small (10x10) board, 100 pegs (20 of each of five colors) and a collection of a smaller and thinner patches to hold the pegs together as they stack up.

The patches make this set particularly unattractive. The designers, probably, wanted to teach shapes. Shapes are the second most popular gimmick favored by American preschool educators.

Not counting the patches, tall-stacker pegs are flawed in many other aspects. They are hard to join even for grownups, and they break. Besides, we simply failed to appreciate those 3D structures. At first, I could suggest an additional layer or two as a finishing touch to make my early "snowflake" layouts look like castles, but I don't think we have ever used the patches for challenges.

Pegs are problematic, but the fundamental design decision was nothing short of brilliant. Being shoe-makers, Lauri had made their board out of crepe rubber, the firm rubber foam of the kind widely used for shoe soles.

I did not like this toy and tried to get rid of it several times, thinking that I had exhausted its challenges. Every time, I simply could not find anything better. Meanwhile, I dreamed up new ideas, and pegboards lived on. Not only did they live - they were branching off new uses like roots.

So I kept ordering more. As of today, we have 700 pegs and four boards, two 10x10 and two 14x14. I have invested over 200 dollars in them.

In the age of "devices", such spendings may seem unjustified. If you really think so, you need a cyberkid. No existing computer interface, no matter how expensive, can match the sensorimotor experience delivered by Lauri's punched shoe sole rubber and the pieces of colorful plastic. You need to see how human children bunch and hug them. No, I would rather buy a plastic printer. I developed pretty clear ideas of what I need from such a toy.

Halfway through the board, I started taking pictures, at first to reward TLG and commemorate her achievements. I printed them and stuck them to the kitchen counter's wall. Just as I ran out of space, TLG lost interest to posing, and I became very interested in documenting our lessons.

I've been an amateur teacher most of my life. Both of my parents were teachers. I have always taught things not being taught at schools, but it was what I learned from other grownups. This time around it was different. Not that my student was making scientific discoveries, of course she was not. She was just doing something I've never heard of.


A record taken out of context to let you know what was going on between us.

That day, before I came up with any challenges, TLG grabbed the reins and ordered a checkerboard. I agreed, but determined to make it worth my while. We did checkerboard patterns a couple times. At first it was hard for her, but then she learned.

I built the layout shown on the left picture below, all at once, and challenged TLG to reproduce it. She boldly took the bait and ended up with the two diagonal lines pictured on the right.

Of course she realized that something went wrong, but what, and where? For several minutes I watched her increasingly chaotic attempts, then offered help, and she was already desperate enough to accept it.

I could have had her build two intersecting lines, yellow and blue, crossing the board from corner to corner. That would have been an easy start. Then, possibly, another failure.

Failures were necessary for the game, but allowing two in one lesson would be an unforgivable mistake.

Trying to save face, I cleaned off my board - "no, no", she screamed, "I want to build this" - and showed an easy way: start from the corner, line up 5 pegs diagonally, make a right turn and finish building the angle. Rotate the board, alternate the color, build another angle, etc. Finally, fill up the angles with the same color pegs.

This was my first attempt to give TLG linear directions, and it worked. The resulting layout was not new to us. Several days earlier I had challenged TLG to replicate it all at once. She agreed and made good progress, but (typical for her) could not finalize the fourth triangle, and left behind several misplacements. I suggested to her to count the pegs in the rows. She became mad at me (guess she thought there must have been an easier way than counting). In the end, I pacified her pointing out her mistakes, and she agreed to use counting to make sure everything was done right.

The cost of the second take was not so high. TLG finished quickly, and I announced the next step: striping.

Striping is a kind of variegating, and variegating is decorating shapes with patterns. We removed four pegs from the yellow and blue triangles and swapped them to create a blue stripe on yellow and a yellow stripe on blue.

Then we removed and transplanted the stripes of three, two and one pegs in one step, and the checkerboard appeared.

If this procedure seems daunting to you, you are not a girl of TLG's age. To me it did. For TLG, such well anchored manipulations were easy-peasy. The surest way to make her mad at that time was to ask her to build a four peg rectangle in the middle of an empty board.

Several days later, we performed a more interesting variegation. Stripes on the following layouts are parallel to the long outer edges of the triangles.

I pictured two pairs of stripes, long and short, being swapped at once. Reproducing this trick, you better handle them separately: a long swap, then a short one. With extra pegs on hands you may put the wrong color in the corner and ruin the project.

Look at what we got in the end.

This snail was not new to us, but the way we built it was. The next week we did it in four colors. It took much more complicated dancing, and, probably, was not so rewarding.

A Brain Twister

Brain twisting is bad for you.

These innocent looking triangles can drive mad a much bigger kid, and even a grownup. Apparently, they meet each other in one central point, like those on the picture, but there is no central point on a 10x10 board.

Had we had such point, we wouldn't have known which color peg to plug in it. The central point cannot be a part of four different shapes having two different colors at the same time. If there is a line separating one shape from another, then again, it can not belong to both of them. The same problem occurs if two lines intersect at some point.

Familiar continuous mathematical plane is even weirder. Suppose you want to exit a blue shape and cross the black line, which separates it from the rest of the world. You may believe you know where you are going to hit the black, but how can you tell where you leave the blue? The shape has no end. Between any blue and black points, there must be another point, and it can only be blue.


By George, she's got it!

Three months later, I offered TLG the crooked cross from the beginning of this chapter. She managed to build it practically on her own, and it was hard. She projected four limbs, but could not get the colors right. Several times she tried to restart from a corner and go across the board (which could have been correct). The same method with the same color for the second time would be incorrect. Well, such was the designer's intention. This commencement was unforgiving. If she made a mistake, she would not be able to fill the triangles.

Eventually, without getting the slightest bit mad, TLG cleaned up her board and built a yellow diagonal line connecting two corners. She checked back and forth, again and again. Yes, I had such. I could hardly refrain from suggesting to her to build the blue line next. I did not realize that she decided to work on one color at a time!

TLG started another yellow diagonal, one hole off the third corner, and counted to four. Another one from the fourth corner, and she's got it. The second and the third lines did not meet in the middle. That's how it was on my board!

The blue part was easy. I thought she would be even happier filling up the triangles, but when I started doing them on my board, she turned mad. She said she wanted to keep this “booteful” layout unspoiled until the next lesson.

TLG's emerging propensity to work on her own until the job was done, and done right, was very welcome. No doubt, school will suppress it, but, hopefully, she will retain something for real life. Through their schools her siblings developed the intellectual responsibleness of beauty pageant contestants. They called out the answers off the top of their heads, no matter how absurd they were, and never bothered to think before opening their mouths. Most remarkably, they expected to get praised just for this. Making sense was not important. Participation was. If I told them their answers were wrong for the third time, they would get seriously upset.

At first I suspected that their teachers were cultivating group ideation techniques like brainstorming, which, presumably, produces solutions out of thin air, without knowledge and thinking. It turned out, the kids were solving problems in groups indeed, but their groups were struggling for smart students. It was not like you learn math facts for 6 and I learn math facts for 7. Instead, they all learned to exploit the few who happened to learn the subject from their parents. And those who did not, they learned to fake participation.

Much later I found that “cooperative learning” was an essential part of so-called reform math. Like everything related to it, it was very wise. "Cooperative learning" can only make teaching easier, and school embraces every easement. Please don't think I am being critical. What would you do if you could define your job?

Then, it's not what you know, it's who you know. Mathematical talents are few and far between. So are unavoidable mathematical problems. Having mathematically enabled friends may very well be all that most American kids will ever need. And if they happen to need, say, math enabled employees, the government will allow them buy more Indians.

The notion that intellectually challenging occupations are for those who can't get a better job is nothing new in the U.S. I could trace it back for at least a century. Well, wishing all the best to their American classmates, I did not think my kids could afford such mentality. They did not even have American parents, to begin with.

Writing this chapter, I was looking at the marble run construction toy. Over a year ago I bought two of them to retire the pegboard. The box promised hours of engineering fun. It lasted a couple hours, maybe. We did not even open one of them. As for the board, those triangles alone could feed my lessons for at least another month.


From dead reckoning to Cartesian plane.

Two months before her fourth birthday, TLG entered the period of rapid development. That's when I started documenting the lessons. The one, which you just read, was the earliest in my records. Counting the rings, TLG had been playing with me for some 20 months.

I am sure you understand that TLG had just learned to count at that time, and that I was exploiting this new ability. In fact, I was not smart enough to do it proactively. She started counting the pegs and the holes on her own. I only seized the opportunity.

TLG figured how to navigate the board in numbers in the course of an entirely different line of challenges. I would gladly tell the story, but it would carry me too far away. In short, TLG did not count to find the center of the board until I taught he to do so. She started counting digging.

Four months forward, and with my amateurish help her mind was quantitized. She understood multitude, magnitude, relations of quantities, learned numerals to 100 and discovered even bigger counts. She was using addition, subtraction, multiplication. She learned to navigate the pegboard through absolute and relative positioning (Cartesian plane and vectors), and even got some idea of accounting.

TLG was smart, but not an extraordinary student. She only learned what a human child could and should learn between the fourth and the fifth years.

Watching his parents, the products of traditional Old World school, trying to help him with reformed North American math, TLG's brother once remarked that we were giant living calculators. Not without disdain, it seemed. He had no idea that I remembered the time when calculator was a human occupation.

Under my persistently bad influence, TLG followed me on this way to extinction. Being only 5 years and 8 months old, she is somewhere between the 2nd and the 6rd grades in math. I can't pinpoint exactly where, because my teaching is very different. I profoundly rewired this subject. I'll tell you more in SHNUMBERS and DEFENSIVE MATH.

When I first read the words "emergent readers" I thought it was a typo. It turns out, the concept of emergent literacy was developed and published by an Australian lady 50 years ago. I did not read her writing, but, according to the other narrators, the big idea was that children learn to read before teachers teach them to. Well, just ask any parent. Granted, the lady was concerned with learning disability.

The notion of emergent readers and writers is very popular in American schools because... let me guess... why bother to teach them, if they emerge anyway? Calligraphy went to the education's graveyard to rest in peace next to the dead languages, followed by cursive, and reading is heading their way because modern kids have YouTube.

The local school district is going to get another proof of concept soon. I spent two month's allowance of my 30 minutes a day and taught TLG to read. She now reads better than her big sister did when she was seven, and she is not alone in preschools.

With due respect, allow me to suggest that there exists a potentially useful phenomenon of emergent numeracy. It could feed another branch of the science of education and liberate a great deal of schools' resources for teaching patterns, number sense, data analysis, art, health, sports, music, cooking and other exciting subjects.


Patterns shmatterns

Three, soon four years since I ordered two sets of the stacking rings, I am still asking myself what did come out of this project, and do I have to make this experience public.

I can clearly hear the fellow American moms saying: no worry dear, you did a great job teaching her patterns and developing her creativity.

Allow me to admit that, after 15 years in your country, the word creativity makes me cringe, and I think your school's obsession with dumbed-down patterns is downright dangerous.

Every person who built a home in a flood zone, assuming that 300-year flood, which happened last year, cannot occur tomorrow, believed in pattern.

So did every person, who blew everything he or she had in a casino or in the stock market.

So did a texting teenager, who killed another motorist.

When I was home-teaching TLG's big sister, a third grader, she was seeing patterns everywhere. It was so depressing that I administered to her a sizable injection of anti-pattern thinking.

As I told you, in the beginning I had no plan and no intentions, except kickstarting TLG to speak. Like her siblings before, she was ready for a conversation. She only needed a party.

A year later I found myself discussing with TLG the strategies, no less. What she did learn I still only tentatively know, but take a look. This heavily compressed barely readable image is a sheet, pulled from our current notebook. Here is what we do every time before a lesson, just to warm her up.

TLG picks a 10-digit number from pi. I pick another one. We have several pages of them printed. TLG dictates me her number, I dictate her mine. She reads them out loud (two billion, one thousand eighty two million etc). She tells me which one is bigger, which one is smaller, and if they are even or odd. She adds them up, often telling at my request what number, even or odd, she expects in each position. She reads the sum. She counts forward and back. I pick a short piece (139 in this case) and she tells me how many ones (139), tens (13) and hundreds (1) there are.

In the bottom left corner there is a chart of additions, which TLG drew. I don't ask her to do it very often because I know she can. I did not teach her "math facts", she easily navigates this two dimensional table, which is just another pegboard to her, and she prefers my diagonal version of it.

She knows long subtraction, but we don't do big numbers every time because it takes her much longer than addition. We only practice subtractions using the chart, and TLG solves them as equations. She learned X when she was four, and she likes it.

Let me stress one more time, everything is perfectly normal here. I taught this stuff to several kids, and I am certainly not the most demanding parent. With TLG I just started earlier, having more time on my hands, and the pegboards.

Why pegboards again? In SHNUMBERS you will see what they can do for arithmetic. But look at the picture: it's all spatial. TLG reads the books and the numbers from left to write. She adds them up from right to left. The chart is in Cartesian order.

It took some work to teach. Like any kid, TLG could not remember what goes where, she was making mistakes, she was getting frustrated, and she learned all this mess more or less at the same time.

Honestly, I am not a school hater. I am too afraid of them, but... may I just tell you something? We quit pegboarding soon after TLG turned four. Three months later she finally went to preschool, and learned two things. She told me that she found out how exactly the boys were different from the girls, and she discovered pattern. Red, blue, red, blue. I understood. I did not use this word with her, and intentionally so.

Could I be fooling myself? What if preschools are right, and a four year old can only get red-blue-red-blue? Well, it was the third American preschool in my life, and I did ask myself this question. The last five weeks of our pegboarding were spent seeking the answer. I built a part of a coarse symmetrical layouts and asked TLG to finish it. Or a fine-grained pattern, which was damaged or distorted, and TLG repaired it. I know what I am talking about.

Can preschool be right in its own way and an average American kid can only get red-blue? I don't believe it, but most of all, I don't care. Even if this was true, I had a child who could do immensely more complicated tasks. It's use it or lose it and, often, now or never. The early development knows no mercy. Teaching TLG red-blue would be debilitating.

Could I forgo school? No, I could not. TLG had every right to grow up American, and school was the only way to go. But I also wanted her to become a full fledged intellectual.

The next year TLG was admitted to the preschool at the local elementary school. No daycare at all, just 4 hours of "teaching" daily. Soon they issued a report card, according to which TLG could count to 40, and this was BIG to them. Per the next report card, it was 60, which was even BIGGER. Meanwhile, given enough time, TLG could count to a trillion or a quadrillion. She dealt with billions every day, and she could go forward and back from any 10 digits number, not only building them, but naming them properly. Don't get me wrong, it not about what I taught, it's about what they teach. They teach 5 years olds counting to 10. To 20 at best.

When TLG's big sister went to the first grade, her reading teacher called me, asking if she could teach her with the bigger kids. She said the big sister was a very advanced reader. This did not preclude the school from enrolling her in ESL lessons. Excuse me, American taxpayers.

In the second grade, her teacher discovered that she could fearlessly add numbers of any length. This did not make the school happy. They wanted her learn "friendly numbers", "math facts", "number sentences" and other stuff, which was making her very afraid.

School wants to teach. The more, the better. That's why this industry thrives on disabilities.




What TLG's background in patterns could have have in common with her consequent learning? Could she just be exceptionally smart? I knew she was not. OK, but could she thrive on parent's attention, like any other kid would do?

I continued to teach TLG searching for the answer. After I had advanced the other projects to the status of demonstrable technologies, I got time to take on this quest. Four years after I had bought the stacking rings, I realized that educators were avoiding to teach young kids anything two-dimensional. My pegboarding was taking place in two dimensions. Could this enable TLG?

To show what two-dimensional thinking could do, I produced an interactive online presentation STEREOLEARNING. I believe, I correctly described and explained some of my findings, but not all of them.

My experience with FIREPEGS had brought about another unifying concept. It was spatiotemporal pattern: a pattern changing and repeating itself in time and space. Normally, if children get to build patterns, they do it according to a static picture. I was siting next to TLG unfolding my challenges in time. For every pattern, I could offer many ways to build it step by step.

Spatiotemporal patterns are immensely important and very little known. As far as I can tell, they are rarely taught and hardly ever used for teaching as abstract forms.

Yet the question lingered: how familiarity with spatiotemporal patterns we built, could help understand, for example, addition in positional notation? It's a spatiotemporal pattern too, but of entirely different kind.

Another year later, I taught TLG, who just turned 7, algorithms. Suddenly, this allowed me to connect the dots.

Small children positively resent the idea of copying patterns row by row or column by column. They want a method or a strategy following what they perceive. This makes the game of MAKE THEM MAD possible and invests it with meaning. MAKE THEM MAD develops an array of skills, poorly understood but very well known to the practitioners of many intellectual activities.

Remember how a fancy cross turned into a checkerboard? We were solving the problem devising the strategies. Both the problem and the solutions were remote from elementary math. However, handling the patterns, TLG employed procedures making repeating steps, breaking the sequences when certain conditions were met, changing parameters or performing the same procedure over a new set of pegs. Needless to say, procedural thinking is exactly what the doctor ordered for learning arithmetic.

Patent pending