Shnumbers teach numbers. Numbers don't.
DEMO CHAPTERS AND SOFTWARE
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One flew over the math salvage yard.
SHNUMBERS is one of those books, which have never been written. I started it from the middle. The time line (found below) shows that during the second Summer vacation I taught a mystery subject named MATH FACTS FOR BIGS AND SMART. That was it.
Next Autumn I drafted and laid to rest few more chapters, transcribing and reinventing the lessons from the first Summer vacation. Then I began schooling the younger girl, learned web-programming, created several software prototypes and, finally, decided to spend some time (an awful lot of it) to put together this demo to see if anybody would care.
Since you are reading this, you, probably, do, even if not how I wished. Thank you for your attention.
Please understand that the original chapters were not intended for WWW. I hastily adapted them from the paper versions.
Another math hater in the family.
TLG and me had been happily playing with the pegboards for six months, then another story started to unfold. TLG's big sister went to school and learned to hate math. She was approaching her Summer vacations, and I was dispatched to help.
Five years earlier this exact thing happened to her brother. Both were fairly regular kids, and I had no reasons to expect any problems at all. By the way, both were good, even better than average at English, and both loved their schools. Their schools were nothing by good and their teachers were doing their job just right. It was their job to make my kids mathematically disabled.
I could not help the boy. His math was not the only problem I was facing, and not even the biggest one. He was more resilient and did not sink as quickly as my next offspring. Above all, I was not prepared.
I did not even know what math facts they were talking about. I suspected, but could not believe that those were just additions. My knee jerk reaction was, get a book. What book? The teacher said they were not using any particular book. Quite clearly, the school didn't want me to meddle in. Much later I learned about home teaching and understood that the teachers were treating elementary math as their professional secret, protecting it from evil parents trying to take away their jobs.
Couple years later, then the catastrophe became undeniable, I wasted hundreds hours of my and my son's time, and failed. He became an accomplished American math hater. I only learned to recognize this mindset.
I remember my failures well, and I don't forgive them. I used every opportunity to understand what's happened. I learned about New Math, The Great Math Wars, Reform Math - all this truly incredible history, what every American parent knows. I noticed that some modern American educators did not like that their school was fabricating math haters, and tried to figure why. They enriched the educational mythology with loosely relevant, but badly misguided (in my eyes) concepts like cardinality and manipulatives. The last one was particularly funny.
Debriefing the girl, I was finding the familiar problems. The sheer amount of strange words and ostensibly important details completely obscured few essential, clear, and unquestionably useful ideas.
She was scared of numbers. She did not understand why she needed them. She was demotivated and demoralized. She told me that math was confusing - can you imagine anything worse than this?
I had plenty of evidence that math was confusing for the teachers too. They did not understand that they taught. Sorry teachers, I did not care if you were heroes or cowards, or just math haters yourselves. I had this child of mine to take care of.
Brain twisting is dangerous.
Please look at the following picture and tell what do all those groups have in common.
Too easy for you? How about your kids? Please excuse me, I only tried it on mine. They failed miserably.
At home I did it with real things. I presented two sets of them and asked this question. I the answer was they are all red, I went on to add something blue. If the answer was they are all made out of plastic, I threw in some metal. Rounded - I had cubes. So it progressed until there was no common property left, except this one.
You still didn't get it? I would be happy to help if I knew how to name this property properly. To me it's a count, but I've never seen an educator using this noun, and I am not a native English speaker. A bit scarier word would be quantity, or, more specifically, multitude. Some preschool educators recently started calling it cardinality, which is absolutely frightening. Two my school kids told me and confirmed that teachers call it amount. That's pretty strange because amount to me is usually uncountable. I took it as another evidence of that clever educational strategy turning normal kids into infinitely teachable dummies (you know, the teacher's job is so hard because math is so confusing).
I thought this easy problem was worth mentioning in this demo because that's where mathematics and math hating start.
SHNUMBERS and the incredible frameless counting frame.
I am now going to explain SHNUMBERS in a few lines. Often, very often we need to pick a quantity and carry it away, leaving the original set of the objects behind. Carrying those elephants just to tell how many of them were there would be not a good idea. To tell how many elephants were there, you would probably say five. Now please remark that five apples or five apple seeds would carry this quantity just as fine.
The process of making a copy of a quantity we call counting. To count we match the objects to some other objects one to one. Those other objects may have absolutely nothing in common with the originals. We simply use them as tokens.
One, two, three, four, five are mental, or imaginary tokens. We memorize them and their order, so they are always with us. Whenever we need to pick a quantity, we match our mental tokens to the objects one to one. Granted, these mental tokens are more advanced than apple seeds or fingers, but they are also trickier to use.
SHNUMBERS is my umbrella word for devices to pick and carry quantities other than that they teach at school. In all fairness, school's numbers are not numbers at all. The word came from Ancient Rome, and them Ancient Romans did not use positional decimal notation. Yet the word number now belongs to school, and there is nothing I can do about it. US school is the most powerful corporation in the world.
When my first American kid grew to become a math hater, I could not exclude that he was one of them, even though our family history and experience left no room for for such an assumption. When the same disaster struck again, I looked up the tools: counting bears, cubes, frames, you know them, they are everywhere. I could only wonder how we parents and our parents had learned positional decimal arithmetic without wasting so much plastic.
I reviewed my purchases and found that I already had the best toy. It was called Lauri's pegboard. That's then I started buying more of them, which, given the size of those oafish pegs, was even bigger waste of plastic.
The reason I liked pegboards, even though I still strongly disliked them, was quite unsophisticated. I realized I could use them as frameless counting frames.
IMPORTANT DISCLAIMER, which I should have made in MAKE THEM MAD. I do not sell Lauri's pegboards and I am not profiting from their sales by anybody else. And I would make them different, if I could.
Counting in SHNUMBERS
For many years, I was a computer professional. With the advent of modern "technology", I gave up. I did not like IBM PC and I deeply mistrusted Microsoft's software. By the time of this writing, I expired. I did not even know a good way to draw positions and colors of pegs on a board. Meanwhile, I needed not just pictures, but moving illustrations.
I made myself learn web-programming (which was not very enjoyable) and fought through my first app. Below are the links to two versions of it. The first (and the most capable one) is for regular computers. The second one is for touch "devices".
Keyboard and mouse
You may pick the one you like, right-click on the link and ask your browser to open a new window. Make yourself comfortable and play around. I'll explain them later.
I am sure you know, but... just in case... it's a positional decimal system using Hindu-Arabic numerals.
Here is how positional decimal system using Hindu-Arabic numerals represents quantities:
2x104 + 0x103 + 5x102 + 3x101 + 9x100
You may replace 104 with 10x10x10x10 if you think it's cuter. To understand addition and multiplication of quantities we need all the power of elementary algebra anyway. If taught properly, positional decimal numeral system alone would provide the greatest learning opportunity. Needless to say, it's not how they teach it at schools.
Because ten is too big.
Let's compare positional numeral system and US currency. Assuming that the number above represents 20539¢, it's monetary equivalent can be, for example
1x$100 + 4x$20 + 5x$5 + 3x10¢ + 1x5¢ + 4x1¢
What's the difference? The money is not positional. I could move 1x$100 to the end because it $100 bill has a face value. If I moved positional 2 to the end, the number would change dramatically. The value of twenty thousand is positional. 2 as such only means 1+1.
Next, the first representation is uniformly and reliably decimal. 1 in the next position on the left is always 10 times bigger. The money is just not so nice. Remark that I avoid the word exponential. Maybe I better don't.
The bills and coins do not even make positional system with variable base like, for example, the one we use for time, because uniqueness of representation is not enforced. Consequently, the total value on hands cannot be easily compared to another value or to the price. We have to count it into regular positional system first.
What you see is what you get.
In this chapters, I discuss the notation, in which the same quantity looks like this
20539 20539 1 o 1 o 1 o 1 o 1 1 o o 1 1 o o 111 ooo 1 111 o ooo 1_111 OR o_ooo
While being positional and decimal. this notation does not use Hindu-Arabic numerals. It uses shnumbers.
Shnumbers can be imaged, like I just did, so they are numerals, but they are much more interesting as the lines of tokens. Later in this demo I will not call them numerals. Whenever I mention numerals, that will be Hindu-Arabic system on numeral names.
Otherwise, the only difference between positional shnumbers and school numbers is that shnumbers do have countable quantity, while Hindu-Arabic numerals only symbolize it.
Positional shnumber can fit any fixed or variable base, provided that we watch and enforce the upper limit or limits for the columns of 1s. In decimal system nothing should be higher then 9. For counting time in minutes we may want two shnumber, one not bigger than 9, another one not bigger than 5.
In lieu of 0 there is a space, which in the computerized word is a valid symbol too. To avoid confusion, space can appear as an underscore sign.
The "tall stacker" pegs, sold with Lauri's pegboards, look very much like 1s, and the stack of them reminds the columns of 1s on the left picture. However, I would not recommend anybody to stack those pegs without a very serious reason. I usually line them up on the board in two dimensions. Looking from above, you see them as shown on the right. The little circles are not zeroes, they are the pegs, and the pegs stand for 1s.
What numbers are natural and what they are.
Let me admit that, like any hardcore computer guy, I firmly believe that there are only two natural numbers: 0 and 1. I never happened to see five elephants naturally turning into elephant five, let alone just five. They always stay 1+1+1+1+1. Two elephant can be different from one elephant because they can make baby elephants. Not to offend anybody, but it's not because of their quantity. The number 2 has never been caught making other numbers.
I am not a mathematician (not even a math teacher I am), but I heard that some mathematicians, widely recognized as such, doubted the validity of the notion of the natural number. Well, this is science. I am only concerned about teaching my kids.
With kids, shnumbers like
do miracles. For example, they add themselves. Suppose you took one apple seed for every one elephant you met, then put them (the seeds) in a sachet and left somewhere. Next day you see 1+1+1+1+1+1+1 elephants. Never mind number facts, you simply match them to the apple seeds - there always is plenty of apples around the elephants - and eventually put your new seeds to the same sachet. Or, if you happened to have your first elephant's count on you, you count forward like
Get it? Good. Now let me tell you something important: in this demo I can not and do not picture and explain every detail. I rely on your knowledge of elementary math and ability to connect the dots. Thank you for understanding this.
The tokens, matched to the counted items one to one, do carry quantity, but they are not a number they teach at school. So, they are a shnumber.
I often say that numbers and shnumbers are the models of quantity. It may be an abuse of the concept of model, but the best word vector (or carrier) is taken. Notation? Shnumbers are not necessarily a notation.
Speaking about models allows me to ask how good those models are, and this is a very productive question. Indeed, a model can be good or bad. Easily. Now, what do we want from a good model of quantity?
From shnumber pile to shnumber line.
First of all, a good model must allow to compare the quantity, which it carries, to the other quantities. Digging a bit deeper, we find what the ability of being compared is practically a definition of quantity. Wherever we meet something, which can be bigger, equal or smaller, we are dealing with a quantity. Matching one to one is the method to compare the countable quantities, also known as multitudes (see, it would be so much easier to call them just counts). Counting is comparing until the quantities of the items and the tokens become equal.
A handful of tokens is not a good model of quantity. Imagine you have the count of horses in appleseeds on one hand and the count of horsemen in wheat grains on the other one. How easy it is to tell if they are equal?
A much better idea would be to use nearly identical tokens, and keep them lined up side by side, or at the same distances from each other. Did you think about a pegboard? It's too short. I was thinking about shnumber line. Historically, it could have been a line of beads.
LIke everything I am telling you about here, I did it with my kids for real. We lined up two big piles of pegs to find out which one was bigger. Not once. That's why they appreciated the idea of folding the shnumber line up and sticking the pegs into the holes to keep them in place.
You know how to get kids do such silly things? Be serious. Gravely serious. Act like if lining up the pegs was a very important business. Because it is.
For the future, let's remember that numbers and shnumbers are to carry quantities. When I got those victims of American Math across the kitchen table from me, they could not answer the easiest and the most fundamental question: what do numbers stand for. With TLG's big sister, I went further and asked her what do letters stand for. She tried on me many answers, non of which was even close. Even when I asked her what people learn first, to speak or to write, she could not tell. TLG was 3 years old at that moment. Eventually, I browsed the web from the home theater PC, found a "phonics song" and played it out. This made my subject very unhappy, but couple months later she did not remember a thing. Then I tried a different strategy, but it's a different story.
Sorry I did not think of it before, but it's two and a half years long!
Since this moment on, I am mixing up two stories, which were over one year apart: how did I "remediate" TLG's big sister, and how did I teach TLG from scratch. They now look the same, but two years ago I was not at all sure if I could use remediation techniques for teaching, and how exactly I was going to apply them. At some moment with TLG I started writing a diary, and this line of events became more or less documented.
|TLG||TLG's big sister|
|SUMMER VACATION ENDS|
|MAKE THEM MAD with rings||1st Grade, indifferent|
|MAKE THEM MAD on boards||1st Grade, emergent American math hater|
|SUMMER VACATION BEGINS|
|MAKE THEM MAD on boards||SHNUMBERS on board (linear counting, positional counting, linear operations, positional addition, positional numeral addition||SUMMER VACATION ENDS|
|MAKE THEM MAD on boards||2nd Grade, declining American math hater|
|WINTER VACATION BEGINS|
|MAKE THEM MAD on boards||SHNUMBERS on board, positional subtraction, positional numeral subtraction.|
|WINTER VACATION ENDS|
|MAKE THEM MAD on boards||2nd Grade, good student|
|SUMMER VACATION BEGINS|
|MAKE THEM MAD on boards, rapid intellectual development||MATH FACTS FOR BIGS AND SMART|
|SUMMER VACATION ENDS|
|Done with MAKE THEM MAD, learns mathematics||3rd Grade, mostly excellent student|
Before the first entry in this table, home teaching was nothing new to me, and shnumbers were even less then nothing. Not only I taught kids quantity on tokens. I also lived through early semiconductor electronics, in which, for example, we could have reliable low voltage and power LEDs, but did not have digital indicators to match.
In the cell, which I highlighted making the word SHNUMBERS bold, I started doing something new: teaching an emergent American math hater shnumbers on Lauri's pegboards.
I am telling you one and the same things several times because I want you to understand. Here is another flash forward: with positional shnumbers and the boards I could teach big quantities before "math facts". To me it was an invaluable opportunity, because only before the face of big quantities all our sophisticated numeracy is useful and makes sense.
Which model of quantity is the best.
I know exactly when did I buy this or that toy set, but don't remember when I grabbed two 10x10 pegboards and graphed them like this.
The left board has shnumber line folded, or, rather, cut into the segments of 10. Imagine a line of 74 pegs, which was transferred to the board peg by peg from the bottom right hole up, then from the second bottom hole from the right, and up, and so on.
The right board has the same quantity, represented as a positional decimal shnumber. I am not avoiding the word positional with kids. They learn tons of words, and can easily remember few more, if they are truly important. TLG knows which board is positional and which is straight, and always corrects me if I use them improperly.
It so happened that TLG's first reading book was Shel Silverstain's "A light in the Attic", and she struggled with "Ations", which I included in my selection.
Details are important, but I can't explain them. Most significantly, I can't tell why did I order the left board from the bottom right hole up, and then to the left. Guess I just decided that there is no way to straighten up the spatial mess of numeracy and literacy, including Cartesian plane, so let TLG learn do deal with it. And she learned.
The straight board can be expanded in 2D or in 3D. I believe, I saw the method in an old book - most likely, it was Robert Recorde's "The Ground of Arts", which I could not find again. Once you filled the board, it's 100. Then you can start filling the imaginary board or boards, holding the boards in the same way as a simple board holds the pegs. If you want to do it in three dimensions, you stack up the full boards or just the pegs - remember, they are "tall stackers" - until you got 10 of them. Then you treat such stacks as pegs.
You are probably familiar with Dienes cubes. I don't believe you are not, but you can always google them up. A peg could stand for a cube, a full board for a square, a pile of 10 boards for a big 1000 cube. Then you can start building sticks, squares, cubes from the thousands.
Kids could walk counting their steps. Now they have to leap.
Meanwhile, I run out of room for a chapter and had to start a new one. It's going to be about the board on the right, which has positional decimal shnumber.
With TLG's big sister, transition from left to right took some dancing around. With TLG I did not even notice it. She definitely learned something from "edutainments", but make no mistake, her sister had all this stuff too. In fact, she had more, but only TLG played MAKE THEM MAD game with me.
The difference in the amount of plastic between the left and the right is striking, and the conversion is very easy. We simply put one peg in the tens position on the right for every whole column in the folded shnumber line on the left. However, I was not interested in teaching this. Instead, we counted the pegs moving them from left to right one by one.
Counting is an operation with quantity, largely ignored by school. Inevitably, they teach some counting, but they don't even consider it to be an operation. In computing, counting up is called increment and counting down is decrement. TLG knows these words very well.
You may think that counting means adding one. Actually, it works exactly the other way around: addition is incrementing many times. Every whole number is a number of ones, and every bit of arithmetic is about incrementing and decrementing. Addition and multiplication are merely the methods to perform counting faster.
With TLG's siblings, my statement that every number is a number of ones knocked off the socks. With time, the big sister came to terms with it but the big brother, apparently, decided that I was deeply delusional, and could only make his situation worse.
I randomly ask TLG one of two questions: what is this number and how many ones are here. The answer is, of course, the same. When we tried positional subtraction for the first time - I don't remember what is was, let's suppose I asked her to subtract 45 from 54, and those were positional shnumbers on the board - TLG claimed that she did not have any more ones. I said, like, what are you talking about, a moment ago you found 54 ones, and you only took away 4 of them. Where are the other 50? Maintaining this grip on reality is quintessential.
Computer can tell more
Remember the app, to which I referred you several chapters earlier? That's a counter. It increments and decrements. You can set up any positional shnumber touching the holes on the board or clicking on them. Click on the question mark to get the help page, which you can also use to turn off the scary parts like the exponential representation. Above the board you have the buttons like on a video remote control. They allow to increment or decrement the shnumber on the board by one, to start continuous incrementing and decrementing or to reach the upper and the lower limits instantly.
Try continuous increment. In this mode, the counter is counting computer timer's ticks. Counting always takes place in space and time. You look and your neurons fire up: they recognized another elephant. That's an event, and some other bunch of your neurons matches it to a number.
If you are watching the counter now, you see how the shnumbers in positions raise and fall, passing accumulated and compacted quantity to the left. When I first thought about using a pegboard for counting, this moment of destruction put me off. I was afraid it would be frustrating. I perceived the little shnumber lines as assets.
I could not have been wronger. TLG's big sister enjoyed getting rid of those buildups and exchanging them for one higher order peg. Clearly, to her they were liabilities. Positional counting on the board instantly became her favorite part of the math lessons. A year and something later, TLG was just as happy to swipe away another nine pegs. On soft Lauri's pegboards, you can do it in one swift stroke.
I found the board better for counting than any other toy I knew. Dienes cubes and counting bears, for example, are not spatially organized. It's hard to keep them ordered. Counting frames are organized, but they are framed. You can not remove the pieces or count from a pile to a wire. On board you do exactly this. In my example, the pegs from the left board went directly to the right board to be used and (mostly) discarded.
The difficulty with the board is that you have to follow specific rules. You don't, for example, mix discarded pegs with the pegs you are counting. Good thing, we don't need to teach the kids to use the board correctly. It's not how they are going to count after they learn.
Money, money, money.
After her first two years at public schools, TLG's big sister became keenly interested in money. I hardly ever touched this subject at home, but her country was talking to her, and I had no part in their conversation. She learned what she could become rich by marrying a rich man. She was finding some aspects of marriage disgusting, but she also knew that with time she would get used to them.
Meanwhile, her natural potency of handling big money was rapidly deteriorating. Simply put, she was terrified with school's numeracy.
She must have been well prepared. We did not rely on ourselves – of course we were misfits – but she graduated from the best available preschool, and in the kindergarten everything was just fine.
I knew it was now or never moment. I had to teach her that numbers are not to confuse us. They enable us to deal with large quantities. Payoffs for the rules and “math facts” (the name sufficed to scare off even me then I first heard it) were enormous advantages, but how could I explain this to a seven years old citizen of the US?
I borrowed a toy pegboard from the lessons I was giving to her little sister, and said – OK, let's try to live without numbers.
It did not take long. Just forty minutes a day for a month. Not even every day, and we talked about many other things too. Like, fore example, celestial navigation.
An untrue story.
The old New England church near us launched “Save Our Steeple” campaign. On the big sheet of cardboard they drew the steeple next to thermometer-style progress bar. The height of the steeple measured in dollars was, I suppose, fifty thousand.
This looked like a perfect opportunity, if only I knew anything about fund raising in this country. Yet I started making it up.
First was the old lady walking her dog. She gave
and the dog wagged its tail.
Next were the girl with her dad on their way to school. They carried
for the school cafeteria. They decided they might as well pay for it tomorrow, so the girl went in and donated what she had for the steeple.
senior tourists gave
each. That was multiplication on the board.
The owner of the gas station across the road put one more jar on the counter and wrote the sigh offering free car wash to everybody who would give
Another lady was running by that morning. She did not have any money on her, but, as soon as she reported to work at the local bank's branch, she told everybody about the fund raiser, so the branch manager offered to match the staff's donations dollar by dollar.
I am cutting the story short, assuming that you see plenty of opportunities yet. Much more than just addition.
We started counting up the donations using the pegs instead of dollars. By the end of the day we were going to update the progress bar, but found ourselves with too many pegs on hands to line them up, and then we ran out of pegs.
Had we have more pegs, what could we do? We started folding the shnumber line, but it was not enough. We could fold already folded shnumber line stacking up the pegboards. That would be 500 of them. Quite a tower. Not as high as the real steeple, but close. Or it could be 50 big Dienes cubes. Stacked up they would reach the top of the second floor. Depending of what we would use as tokens, we might need a big bag or a big wheel barrow just to carry 50,000 of them around.
Folding or pleating shnumber line may help to handle big amounts until they get too big, and too big in case of $50,000 church steeple renovation project was not big at all. Median home price in the area was around $400,000 at that time. Boeing Dreamliner's cost was over $200 million, not to mention $32 billion to create it, as of the last available report.
Obviously, we were lacking the right tool for the job. We could obtain more pegs, but only to get buried under them.
I briefly reminded that the renovation of the church steeple, even though it was the biggest attraction of this charmingly unremarkable village, was not the most expensive project the people have ever taken on, and suggested to call power tokens to the rescue.
Positional system, counting and addition.
Power tokens are tokens of tokens. The idea may be hard to get at first, but it well worth trying because it's very powerful, indeed. If token can stand for practically anything, be it an apple, an elephant, a student or a class of 25 students, why can't a single power token stand for the several plain tokens? It sure can, and it would be a power one token. Then, a single power two token would stand for several power one tokens, and so forth.
I hope I explained enough. In course of several thought experiments I brought my student face to face with big quantities. The biggest questions had been answered. I could do it because I detached clear and beautiful exponential positional system from the toxic mess of math facts.
I threaded very carefully with her, given her emotional conditions, and we took every step from shnumber line to positional decimal Hindu-Arabic monster. In the end she was fully trained to count any quantities and add long positional decimal Hindu-Arabic numbers. Speaking of fully trained, I meant I got her through scary and misleading numeral patterns, and I could not find anything else, which would stop her steady progress from right to left. She was not confident with "math facts" yet, so what? Learning them adding big numbers is much more fun.
Next Autumn, the teacher send us a letter telling what to expect: addition of one-digit numbers and slow advancing toward two-digits. That's after we stopped just short of subtracting long numbers.
The school went on, teaching "number sense" and zillions scary little things supposed to help the helpless little kids to learn to do positional addition in tests. Several months into the year the teacher discovered that TLG's big sister knew how to add the numbers of any length, and was not afraid to do this. The teacher was kind enough to remark her observation, and even to put it to our attention, not being happy with her overall achievements yet. She still did not meet their requirements.
It took more time for the seeds to germinate. Finally, she started understanding that school was plain silly. Her performance is currently excellent.
Treating one, two, three digit numbers as a different entities, requiring specific knowledge to deal with, is, of course, very beneficial for the industry of education. Guess if they taught to fasten up the shirts, they would do it one button a year.
Everything I did to get TLG from shnumber line to positional shnumbers in one long chapter.
In MTM I told you that TLG knew how to count to 100 then she was about 4 years old. What else could you expect from a child using 100 holes practically every day? I only drew the numbers next to them. She easily learned the numerals and the digits, and more or less steadily remembered them since then. Like any child, she could have memory lapses then she learned something new (I do too). The board was always with us, and we re-learned.
Positional decimal numbers, which we know from school, are very helpful with large quantities. Unfortunately, the educators are afraid of anything large. That's why their teaching makes no sense for the kids. Another trouble is, the educators are trying to teach too much and too quickly (ending up teaching nothing infinitely slowly).
I knew this, but US school was a riddle I've never seen before and have never expected to see. Among other things, after TLG's big sister traveled down the sink and back, I've got pretty clear idea why my kids were so unfortunate with math. That's why I started thinking about DEFENSIVE MATH. I could not go on the offensive. I would hate myself if I did.
I understood one more thing: I had to teach TLG numeracy before US school would ever touch this subject, or I would have to raise another American math hater. It was hard to swallow because I've never wanted early development for my kids. Ambitious parents can teach their kids practically anything. I just have never had such ambitions.
So we were drifting away from shnumber line to positional shnumbers, talking about their advantages and disadvantages. The linear model seemed simpler at first. A three year old could build it counting. Positional shnumbers were not so immediate. Instead of moving to the next column on the board, you stay in place, clean it up and forward a power peg to the left. Good thing, you don't need eraser to change a shnumber. You don't have to remember a carry too. You just add it up, and if it incurs another carry, you perform it at once. Look again how the counting app does it.
For everything besides counting, shnumber line is not so great. The cost of creating and owning it is high, and it grows neck no neck with the quantity it represent. So is the cost of copying, transmitting and reproducing. Addition - which means, counting forward by a given shnumber - is simple, but not easy. Compare sticking another 100 pegs in the holes to adding 100 positionally.
The biggest problem of early math is: kids have nothing to count. It's not at all unique. As another example, we teach them walk, and then they have nowhere to go. I am just not talking about walking in this demo.
In our lessons with TLG, we've never counted more than 140 items. I only have two 10x10 pegboards, two 14x14 boards and 700 pegs, which is, 140 of every of 5 color. We keep every color in a separate big plastic bag and watch them, but they run away and hide in the most unexpected places. So it was perfectly justified to ask if we had more yellow pegs or red pegs, and line them up to find out. When folded, 140 pegs fill a 10x14 rectangle on a big board.
Playing MAKE THEM MAD, I used every opportunity to teach TLG quantities. Once I appointed myself the pegs keeper. I kept all the bags at me and handed pegs to TLG at her request: how many reds, how many yellows, etc. She was building polyominoes and big chunks of the symmetries at that time, so she had to plan ahead. Not sure if you appreciated this enough. To count the pegs, scattered across the board or organized in awkward shapes, TLG needed a strategy. A pattern of multicolor spinners is not a folded shnumber line.
Having finished building, we took layouts apart to count how many pegs went into them, color by color. Pegs patterns provided material for multiplication and division, and we did this too.
I taught TLG to count from any positional number using books. I could open a book at a random page and ask, what would be the next and the previous page numbers. A year later Shel Silverstein's book of short poems came in quite handy.
At some point, I put before TLG three versions of 37: as a folded shnumber line, as a positional shnumber and as a number in our regular Hindu-Arabic notation. I asked which one did she like most. She pointed at the last one. I asked her to add 8 to it. She could easily do this on the boards, and had no idea how to add on paper. I realized that the stage was set. We reached the point of no return.
First, I introduced her to the chart of addition and taught her to draw it. I let her forget it once and re-learn. Several months later I printed out the first page of the numbers from the endless tail of pi. TLG was not happy to see them, to say the least. She never liked to learn anything new. Yet, on the following day she agreed to pick ten digits group and dictate it to me from right to left. It was hard because she started reading by that time.
But positional does not have to be decimal.
I was afraid to tell you of one thought, which crossed my mind when I was considering Lauri's pegborad as a counting tool, but I already did. Unlike any other positional toy, it does not have to be decimal. To put it properly, it does not enforce decimality like a regular counting frame or Dienes cubes.
In the counting apps you have a Hindu-Arabic-alphabetic numeral indicator, displaying the shnumber on the board in human readable fashion, so to speak. You also have two controls for so-called bases. Base is the number at which we stop counting, zero the position and carry. In the system, which we know from school, it's ten. The only reason for it is, probably, the fact, that we human usually have ten fingers.
The counters don't care what did your school teach. They can count to any available base and display the same quantity in any other available base. I limited the list of the bases to keep it manageable.
In the English speaking world, people still use non-decimal positional systems, most prominently duodecimal, or base-12. In the part of the Western world, which followed Napoleon's reform, the only non-decimal remnant is time. However, there are countless ad-hoc enumerations, typically using mixed bases. They play very important role in information sciences and practically everywhere else. In computer programming, for example, such enumerations are produced by nested loops.
I wanted non-decimal bases for their educational value rather than for any practical reasons. Those who know only decimal notation simply cannot understand it.
This tens, hundreds and thousands are unbreakable.
Dienes cubes were invented by Hungarian gentlemen Zoltan Paul Dienes. To prevent you from struggling with his last name (and to the best of my knowledge), the second letter i makes the sound like ea in eat, but very short, and the third letter e make the sound like ye in yellow. Think Diego. The trailing s sounds like sh.
I don't know what Dienes did with his invention. I use the cubes for reality checks when I hear particularly outrageous answers. For this reason, or because they are so scratchy, the sisters hate them. TLG can get mad at me for the rest of the day if I ask her to bring the cubes to the table. Thank you Dienes, you created a wonder of the scares.
Dienes is also credited for inventing so-called multiple embodiment principle. Again, I don't know what did he write about it in his books. I only know how the modern American teachers understand it. For them, the more distractive and potentially destructive ways to teach one and the same scrap of math they introduce, the better they do their jobs.
In search of yet another embodiment, some educators demonstrate knockdown fits of creativity. One preschool teacher told on her website that she does some kind of "data analysis" (which is another gimmick to me) making the kids classify the gingerbread men in the macabre table, drawn on the floor, according to the body part, which was bitten off, and then count those casualties by their injuries. I would not recommend anybody to do this. Food - especially sweets - is for the kids what sex is to their parents. They shut down every intellectual faculty, chewing a gingerbread man's head or leg.
To help you appreciate Dienes' teaching, let me remind you that usually we learn mathematics the other way around. We learn - I don't know what did you learn - let's say, quadratic equation, as an abstract device, and only then reduce it to practice.
Positional does not have to be decimal.
I believe, the positional notations with different bases is the best possible multiple embodiment. Base-4, base-5 or base-6 are also much better than the conventional decimal system because carries or borrows occur much more often, and I can fit two low shnumbers into the board.
Kids who learned that decimal system is not alone before wrapping all their brain circuitry around it, enthusiastically count, add and subtract non-decimal numbers. The boards lend themselves to converting the quantities between the different bases through direct counting.
I would not teach the kids non-decimal charts of addition, though. TLG's big sister built some of them, and I believe it was good for her, but only after she learned the one, which school wanted.
Changing the base of the board is as easy as drawing a line, at which positional counting must stop. Nothing else changes. You learn and run every operation in exactly the same way. If you have enough headroom (and you are not counting shnumbers using Hindu-Arabic numerals or their English names), you may not notice a difference between, say, base-9 and base-10.
The counting app is built around a single quantity, which, at any moment, is displayed on the board as a shnumber, and on the indicator as a number. Change the base, and the display changes. The other display stays the same, so you can convert the quantities between the bases. Remember though: different bases have different capacity. For example, a decimal (base-10) number bigger than 1023 won't fit a 10-digits binary display.
Kids, who learn arithmetic and algebra without math facts, can see math facts differently.
In the beginning of this part, I reviewed addition facts in two chapters. The first one was named FACTS OR FICTION, and I tried to explain why did I dislike the idea of learning the elementary additions as facts. The second one was WHY CAN'T WE ADD, and I tried to explain why do we need to learn the elementary additions at all.
I decided to keep in this demo only the third chapter because it shows how education pushes numeracy down the throats. I am exaggerating, I am understating.
The biggest elementary school secret revealed.
I promised to avoid finger pointing at all costs, but this was simply too much. Today is May, 27 2013 and I just checked with the online service, which school made us use every day. The domain name is xtramath.org. Not much else is known about it. The operators do not tell, for example, who sponsors them. But I just wanted to confirm how do they represent additions.
You don't need to be a member. They have a movie proudly demonstrating their interface. On the following picture the green square on the left is what I found for addition “facts”. They call this contraption the Mastery Matrix.
UPDATE: Today is June, 25 2015. Two years later I found the same movie and, evidently, the same technology.
Hidden behind the green tiles is a regular chart of addition, which starts at the upper left corner. You are not allowed to see it though. They show a single addition if you hover the cursor over a tile for few seconds. The color of tiles depends on the student's previous answers. My table was all green because they thought TLG's big sis was flawless at that time (to me she was not).
Only the parents are entitled to see the numberless rectangle of colors. The kids are presented with the pairs of numbers popping out of nowhere. They must enter the right answer quickly, period. They are reliably prevented from figuring any connections.
On the right, I revealed that strictly guarded trade secret: the same table of addition in its naive form. Blue numbers were added to the red numbers and the answer for every pair is found on the intersection. I will suggest your students to map their knowledge on their own using such chart.
That's what I don't like about “math facts”. The word implies that additions are separate unrelated pieces of information, and their results can only be remembered.
You may say that this is what the website is about. They simply imitate flash cards. But why don't they let the kids to put their answers in context, and why do they take such a great care to blindside the parents? Instead of using the power of computer to teach, they use it to obscure the knowledge. I wonder if the teachers are allowed to see the chart.
I want my and your students to see the whole little big picture, to understand how operation works and to use this understanding to reduce their reliance on ever failing memory. Familiarity with the space of addition will let quickly find forgotten (or never learned) results at any moment in the future - and to repair their mental chart. This is that I would call mastery.
Flashcards are very useful to memorize the facts, if we can only do it by rote, or if we use and refresh this knowledge often. They may help to prepare for a test or for a fight. Teaching the modern kids additions without even letting them see how they are related to each other, is a way to disable, and nothing else. Kids will quickly forget everything they learned - I know it, I did such tests - and they will retain little or no lasting knowledge.
By the way, using a free spreadsheet program any parent can create a random questions generator in half hour, and teach the child quite a bit in the process.
I spent most of my life in USSR, which in the Western world was chastised for manipulating information, yet I could not think of a more striking example of such manipulation from my past. And I don't believe this is accidental. On the contrary, xtramath.org accurately reflects what I see in modern elementary education: they kill the knowledge, drain off its blood, draw the guts, push the soft tissues through a sieve leaving the bones behind, stick on bureaucratic labels and feed this stuff to stupid, paranoid kids, unable to chew and swallow the real thing.
No child left behind - OK, but how snowballing of concepts, terms, details may help those constantly falling behind? As far as I can tell, it does not. For very smart kids, smart enough to challenge taboos, school may be harmless, just because all it's inventions are ultimately silly. The regular children, perfectly capable of learning live mathematics, being forced to consume this mental food can not digest it.
With time they get used to school's diet and learn to do what the teachers want. They learn something, yet to me their mechanically separated knowledge does not even smell like the original.
The whole this part about "math facts" I wrote because I've got a very reasonable and completely idiotic vision: positional decimal numeracy with Hindu-Arabic numbers, which is thought to be a prerequisite for further education, just isn't.
The friendly devices like pegboards - I envisioned more of them, and even programmed some - arithmetic and elementary algebra. The Hindu-Arabic numerals just drag alone. Every kids already know them, and they are very handy. They also require "math facts", but "math facts" are not really worth teaching. TLG is learning them adding big numbers. She is proficient with my diagonal chart (she re-learned it several times). Currently, I encourage her not to use it.
Positional decimal arithmetic with numerals can be explained to the kids who already understood it in all it's beauty: with exponents (remember power tokens), algebraic tricks, etc. Working with TLG's big sis, I thought it would be in the third grade.
It did happen. She learned exponents and distributivity (incredibly, she remembered it - she could even even crack (a+b)x(c+d)) - and I taught her positional multiplication the right way. She could use the dreaded "algorithm" when she was a 3rd grader.
TLG had learned all this stuff and started performing long multiplications before she turned 7.
This made me worry about elementary additions, which American school like to call "facts". I didn't see any practical reasons to teach them. TLG's big sis, after I shnumbered her, became the best in class on xtramath.org.
Meanwhile, math facts are very attractive subject of its own, and children can be motivated to learn more about them as soon as they understand their role. The chart of addition to me is a miraculous piece of laboratory equipment.
Having shnumbered TLG's big sister for the first time, I started making pictures and drafting lessons in Libre Office preparing for the second Summer vacation. The material was not for the web, but, maybe, for a book. However, as I moved forward, I understood that hardly anybody could be interested in such a book.
School is in control. People only want in please it. The idea that kids can learn something that teachers do not teach, and end up knowing that they teach, must be, I'm afraid, just too much for most.
Meanwhile, school, as a corporation, is, and will always be hostile to such attempts. At any moment, school is a bunch of fresh minds, who can learn anything. You can imagine XVth century students in a modern school. They may do just fine. Examples are abundant. However, school is also a bunch of teachers.
Yet I decided to include few chapters from MATH FACTS FOR BIGS AND SMART in this demo. Hope at least somebody will find them useful or interesting.
Finally, a whole new dimension in learning!
Every square in the chart of addition has three numbers associated with it. Red and blue numbers show how to find this square. Black number tells the sum of the red and the blue addends. Using those numbers as latitude, longitude and elevation I turned a flat chart into the Mount of Addition.
On three-dimensional Mount of Addition every result is raised to it's height. Or you can imagine that this is a three-dimensional chart, made out of cubes instead of squares, and that every result is represented as a stack of those cubes. Below 18 are 18 cubes, etc.
The Mount of Addition is a shnumbers thing. Cubes are tokens, stacks are shnumber lines. Only the top stickers with have Hindu-Arabic numerals.
Sorry for the pictures. I made them over 2 years ago for the second Summer SHNUMBERS camp with TLG's big sister. The program was Draw from Libre Office, the pictures were for home printing, and I've never intended to publish them on the web. I currently use InkScape, I just don't feel concerned enough to redraw.
Ugly as they are, the picture demonstrate and explain some weird features of the chart of addition. You can see how exactly and why do we have to jump by two going up in every direction parallel to the line connecting 0 and 18.
We can also walk through the equal results, but it's not easy. Indeed, we have to add and subtract 1 every time. The easiest and the surest ways are always plus-minus ones.
The Mount of Addition provides ample material for teaching. I spent several more chapters playing on it's slope, mostly explaining how do we, the giant living calculators of the past, fill the memory holes and find that we've never learned. I realized, for example, that I did not know additions to 9, although I performed tons of computations by hand, and never in my life owned an electronic calculator.
OK, I bought one calculator at approximately the same time when I was teaching SHNUMBERS. It was a Sharp's model featuring a "number fact" quiz. They provided a reference to a science article in the manual. Somebody bothered to prove that people are forgetting math facts as they grew older, which was billed as a corollary of the mental degradation. The funny moral was, buy this calculator and do the quiz to stay mentally fit. Much more important for me was the metanarrative: we go to school, we get programmed, and then we spoil.
When I started teaching my American kids, I discovered that I forgot how did my school taught me to perform long multiplication. I had no trouble doing it, nonetheless, because I knew the basics of positional system and, of course, algebra. In a day or two I reinvented the method, but I was afraid to teach it, particularly, because I did not know how do they handle carries in American schools.
It turned out, American schools were no longer sure how to carry right, so I ordered a video course, in which a university professor was teaching college students elementary math. Well, he was using eraser. My next question would be how do they check their calculations for mistakes, but I already knew this was hopeless.
Please excuse me this digression. I am getting back to business. When I see something like the Mount of Addition, I want to build it. I actually started building them on the board, and yes, it was a bamboo grove. Fortunately, halfway between base-6 and base-7 I ran out of pegs, and only then I asked myself how many tokens does it take.
tl;dr Don't mess with your brain.
Let start from the summit and count by the layers. The label for a layer will be the visible number – 18 for the highest, then 17, and so on.
18: 1 17: 1+2=3 16: 1+2+3=6 15: 1+2+3+4=10 14: 1+2+3+4+5=15 13: 1+2+3+4+5+6=21 12: 1+2+3+4+5+6+7=28 11: 1+2+3+4+5+6+7+8=36 10: 1+2+3+4+5+6+7+8+9=45 9: 1+2+3+4+5+6+7+8+9+10=55
Until this moment we had a series, known as triangular numbers, but now the rule has changed and we are counting back.
8: 1+2+3+4+5+6+7+8+9+10+9=64 7: 1+2+3+4+5+6+7+8+9+10+9+8=72 6: 1+2+3+4+5+6+7+8+9+10+9+8+7=79 5: 1+2+3+4+5+6+7+8+9+10+9+8+7+6=85 4: 1+2+3+4+5+6+7+8+9+10+9+8+7+6+5=90 3: 1+2+3+4+5+6+7+8+9+10+9+8+7+6+5+4=94 2: 1+2+3+4+5+6+7+8+9+10+9+8+7+6+5+4+3=97 1: 1+2+3+4+5+6+7+8+9+10+9+8+7+6+5+4+3+2=99
Don't hate me please. I know, not everybody likes such things. By the way, I don't. While your student is practicing addition, I am telling you how to get the answer easily.
Imagine a copy of the Mount, flip it upside down and turn its back to you. Put the two bodies together, and they will fill the green cage on the picture. The size of the cage is 10x10x18 cubes. One half of it is 900.
This is the power of positional arithmetic. We could tediously build the Mount of Addition counting every cube (I keep telling you, all we need is to be able to add ones). Or we can get instant answer using multiplication and division.
Now here is one of those things that elementary algebra can do for us. The numbers of the cubes in the layers,
by commutativity and associativity, can be presented as
Building the whole Mount from cubes may be too challenging, but it won't be in smaller base - for example, in quinary (base-5) system. Quinary Mount of Addition is everything from 0 to 4s (both ways), and then to 8. How many cubes would you need?
The following picture shows the Mount of Addition dissected. Surprising, isn't it? It appears so symmetrical.
Explaining how and why would be like couple chapters too many. I did not invent this thing, I believe I saw it somewhere, and as I started playing with it, I was surprised. That was one of the reasons I decided to write a part entirely about addition "facts".
There is a great deal of misunderstanding of arithmetic nowadays. Many people think that in the age of the portable computers it"s no longer needed. Somehow this reasoning - or, rather, sentiment - gets directed toward "mathematical algorithms".
Different people may inhabit different realities. In the part of the Universe, where I live, you may no longer need "math facts" - well, as long as iPhone is on you. The smarter are the phones, the dumber are the users, and being dumb is so enjoyable, you know.
The "algorithms" is that I demonstrate and teach on shnumbers apart from "math facts", and they are everywhere. They are all-pervasive, if not invasive. In SOFTPEGS, for example, you can learn that positional system is a bunch of nested loops, and loops made programming possible.
I could build a theme park around the Mount of Addition, but I doubt if I will ever have such an opportunity. At home you can build the full decimal model or any part of it from cubes, beads or balls. Big cubes may stand on their own. To hold small cubes and spheres, construct a support. Essentially, you need a square hemi-cuboid – three faces of containing box sized as 10x10x18 – put on its vertex and held in this position. For spheres it can even be cut out of a paper box, although transparent material would be the best. For cubes any inner curvature in folds would be a problem. The best solution I can think of would consist of three triangles with flaps folded and glued outside. Scratch the folding lines with a ballpoint pen (I am sure you know how to do this).
Could we add without a board and the facts? The answer is very simple: just count forward. I considered such possibility in several chapters, not included in this demo. Included are only three related to the chart of addition, which, really, is a count forward table.
At some moment, I started thinking about teaching positional numeral addition before learning any math facts at all, using the chart. I was not sure, however, how hard can it be to teach a child to use a two-dimensional table. A year later with TLG I finally found that it was very easy.
Unfortunately, the chart is a device to be carried around. Or... wait, it can be drawn on paper as soon as a student leaned to write and to count to 18, and we use paper to perform long addition anyway.
The only problem left was that drawing the chart takes time. The little TLG could spent the whole lesson working on it. So I embarked on completely insane quest to make such technology feasible. It was insane not because the idea was unpractical, but because every sane student would have learned math facts drawing the chart again and again. TLG's big sis quickly did just this, and, probably, doesn't remember the optimized chart, which I made her draw every day.
Yet I wrote the following, if only for the heck of it. Just beware that there were several chapters before this one. One of them was named TOY ALGEBRA.
How to add without a board and the facts.
Remember why I started talking algebra? We can now return to the chart, to eliminate those upside down entries. As long as we can find that 7 and 6 make 13, we don't need to look up the same numbers as 6 first, 7 next. We are smart enough to always start with the bigger number.
Here is what was left.
9 10 11 12 13 14 15 16 17 18 8 9 10 11 12 13 14 15 16 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 5 6 7 8 9 10 4 5 6 7 8 3 4 5 6 2 3 4 1 2 0 1 2 3 4 5 6 7 8 9
I suggest two more improvements. First, let's move the bottom line to the top. We may shuffle rows and columns as we like, and still get what we want.
tl;dr Don't do it.
The table below looks like a total mess, yet it returns perfectly correct answers.
6 9 7 10 11 15 14 13 12 8 9 12 10 13 14 18 17 16 15 11 7 10 8 11 12 16 15 14 13 9 5 8 6 9 10 14 13 12 11 7 3 6 4 7 8 12 11 10 9 5 1 4 2 5 6 10 9 8 7 3 8 11 9 12 13 17 16 15 14 10 2 5 3 6 7 11 10 9 8 4 4 7 5 8 9 13 12 11 10 6 0 3 1 4 5 9 8 7 6 2
This is not a counting up table. It's like the random questions from extramath.org: you can see only the addition, which you are looking for. For a regular math facts sufferer it must be just fine. A child who learned to use the ordered chart can get mad, but I find it useful to practice with such tables every once in a while to enforce Cartesian thinking. TLG tends to slip to counting forward, which is easier.
Note that I did not randomize it, I just transposed some rows and columns by hand. The general pattern of small numbers in the bottom left corner is still clearly visible.
Where was I? Suggesting to move the bottom line to the top? Think of a number wheel. It counts from 1 to 9 and return to 0. Like the numbers in the leftmost column.
0 1 2 3 4 5 6 7 8 9 9 10 11 12 13 14 15 16 17 18 8 9 10 11 12 13 14 15 16 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 5 6 7 8 9 10 4 5 6 7 8 3 4 5 6 2 3 4 1 2
Using this table, start from the lower left corner and go up looking for the bigger addend. The answer is in this row. Once you found it, continue to 0, turn to the right and look for the smaller one. The answer is in this column.
The second improvement: drop the small numbers. Guess, most students must be comfortable with everything below 6. If so, our addition facts boil down to the following.
0 1 2 3 4 5 6 7 8 9 9 10 11 12 13 14 15 16 17 18 8 9 10 11 12 13 14 15 16 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12
How so few numbers could have been causing so much troubles to so many kids for so many years? And we only have eliminated the unneeded parts. There also are ways to compress the chart, like a modern photo camera compresses the images, but without any loss of information.
The top line, by the way, is rather good to have, than necessary. Instead of using it, the operator may just count up (not forward!) to the second number. Suppose you need to add 7 to 8. You look up 8 starting from the bottom and going up, then you count to the 7th place on the right. This can make a useful exercise, but must be slower and less reliable than looking up the second addend and finding the intersection.
Making this table on paper by hand, start in the lower left corner from the smallest number you student still needs to learn (I assumed 6). Go up counting and writing down every number until you reach 9. Turn to the right and keep doing it until you reach 18.
9 10 11 12 13 14 15 16 17 18 8 7 6
The row starting from 9 contains the biggest numbers. Filling it first you allocate enough room for the columns. Now add the top index row (if you are using it).
0 1 2 3 4 5 6 7 8 9 9 10 11 12 13 14 15 16 17 18 8 7 6
Finally, fill up the remaining rows, counting from the first number and stopping at the diagonal, where the numbers are getting added to themselves.
0 1 2 3 4 5 6 7 8 9 9 10 11 12 13 14 15 16 17 18 8 9 10 11 12 13 14 15 16 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12
Let me repeat, do the bold numbers first. They form the groundwork. Everything else is aligned according to them.
Creating this cheat sheet by hand just took me 45 seconds. What? No, I am doing this for living. With little practice you can easily surpass me. And don't keep those tables. Making them is the way to learn. The more, the better.
Once your student gets comfortable with the cheat sheet (plus, I assume, learned to handle addition in positional system with shnumbers), he or she knows how to swim. Big numbers are everywhere. Open the gates, let them pour in. Your income, your mortgage, your cars, your city's budget, your school's costs, federal debt – sky is the limit.
If you don't want to cheat (I hate doing this too, unless it is customary or absolutely necessary) you may use cheat sheet as a warm-up exercise. Have your student draw it, look at it and throw it away. As my experience suggests, it's more of a teaching, than of a cheating tool. Most of all, it helps boosting self-confidence.
I feel misgiving about who is cheating whom though. Paper and pen are calculating tools. Not just facilitating, but actually working, as you could see. If they are available and allowed, there are no good reasons not to use them to the full extent. And allowed they are, unless schools administer mental calculation tests. Well, never mind. I am not old enough to remember mental calculations at schools.
Let them solve it.
The picture below is from Russia's Moscow Tretyakov Gallery. The year is 1895. The teacher is a former Moscow University professor, who chose to establish a boarding school for the sons of recently liberated serfs (or slaves). Their mental capacities, by the way, were seriously doubted, even though they looked not much different from their owners. Remark the footwear weaved out of basswood bark, and, obviously, well worn by their parents. The artist once was a students at this school. He might have pictured himself.
Here is the problem they are trying so hard to solve:
Not a big deal, but can you solve it without writing?
The official American teachers' recommendation, according to my kids, is: in case you forget a math fact, use number line. The cheat sheet is a collection of number lines, which gives you answer much quicker and teaches you additions as you look them up, simply because you see them all at once.
Still the goal is to memorize those additions, and, eventually, to be able to add without any tools at all, because next our students are going to learn multiplication.
I understand it's not enough, but I am not sure if anybody will be interested to read even this. The market doesn't want this stuff. Wherever I happened to live, I've found several horse farms, dance studios, martial arts classes in the immediate vicinity. Only once I saw an extracurricular math teaching business. In one year they reduced themselves to tutoring, and disappeared from the landscape.