20160805 SMARTING UP:

Clever Hans and Bright John

Seriously, how can they not understand that those Harlem Shake videos are metaphors of ejaculation? Are they pretending? OK, that's none of my business. I just wanted to check on so-called Internet memes, specifically on one of them, which states that multiplication is not repeated addition.

When I came across this idea, I thought that it belonged to Defensive Math. Yet, upon closer examination, I saw familiar logic, according to which fingers are for nose-picking. If multiplication is repeated addition, who needs school to teach it?

Soon I found that the meme was being promoted to educators, and it was gaining traction among them. Now, what is good for educators is rarely good for students. Defensive math is about defending students from teachers, not the other way around.

As a parent, I thought MINRA can be very damaging, and it could explains some peculiarities of elementary school teaching. Otherwise, I did not find MINRA apologetics remarkable, except the following point. Here is the link and the quote.

"Oh, so multiplication of fractions is a DIFFERENT kind of multiplication, is it?" a bright kid will say, wondering how many more times you are going to switch the rules. No wonder so many people end up thinking mathematics is just a bunch of arbitrary, illogical rules that cannot be figured out but simply have to be learned - only for them to have the rug pulled from under them when the rule they just learned is replaced by some other (seemingly) arbitrary, illogical rule.

Irony? Hardly. The author just wanted to make such bright kids happy.

Folklore has many tales of bright guys and the rugs pulled out from under their feet. The number in Aarne-Thompson classification is 1696 (The stupid man/What Should I Have Said?). Here is a collection, thank you professor Ashliman! The first one was retold by Brothers Grimm. They included another one in their book. It's named "Clever Hans". Ironically, of course.

Links are unreliable, so please allow me another quote from Wikipedia. It's short and more understandable, although, I suspect, is not free from mistakes. For example, why say "not much"? Or - from another story - why put a knife and how to put a goat in the pocket, unless it's a pocket goat? Was it a bag of some kind?

A long long time ago there lived a mother and her son. The son told the mother that he wanted to go out a traveling. The mother was very worried about it since they were very poor. The son told her that he would be fine, and he would always say "not much". One day on his travels, he passed by a group of fishermen while he was saying "not much". The fishermen could not catch any fish and were very angry at him. He asked them what he should be saying instead. They told him to say "Get it full". He continued to say "get it full, get it full" while he was traveling. Then he passed by a gallows when some prisoners were being hung. The executioner got angry and said, "so it is good to have more criminals?". The young man asked what he should be saying instead. The executioner told him to say "God, please have pity on the poor soul". Then he came across a group of knackers who were skinning a horse while he was saying "God, please have pity on the poor soul." The knackers got mad and told him to say "there lies the dead flesh in the pit". So the young man kept on traveling while he was saying "there lies the dead flesh in the pit." A cart passed by and fell into a pit. The people in the cart were mad and start attacking the young man. He ran back home and never went out a traveling again in his life.

The story is clearly not for a school library. Synopsis failed to mention that everybody ended up "attacking" the stupid one, hurting him pretty badly. I gave Brother Grimm's stories to TLG to read. Another reason to wait for police to knock on the door.

Comparing two quotes, I can't help but marveling at the cultural shift, which mass education induced in the last 250 years. Dumbing down to cater to imaginary masses they said? The tales collected by Brothers Grimm were not exactly pinnacles of intellectuality, but their Clever Hans is now our Bright John.

I wonder what even moderately dim student would fail to understand that math is a developing knowledge, and when you learn math, it develops in your mind. They tell you one-two-three-four-five-six-seven-eight-nine-ten are numbers. Then you have to go to 20. And zero. And -1. And fractions... and now I feel like the rug is pulling my feet. Why, why, why do they say that multiplication of fractions is not the same multiplication, which has been known for millenia?

I can't speak for the masses. They can be this stupid, of course. I can also tell that the masses are being taught precisely this kind of mechanically separated math, in which nothing makes sense even for the teachers. Another factor is obvious too: stupid students are good for the schools. First, the teachers themselves don't have to be smart. Then, nobody can expect and demand the stupid students to learn much of anything. US educators grew very good at picturing their students as stupid, making them stupid and silencing those who don't accept this role.

By the way, the first quotation only makes sense if Bright John will live to learn true math. This is then the false knowledge of multiplication as repeated addition will return to bite his butt.

20160730 SET THE STAGE:

What does it take to multiply?

Two weeks ago me and my still-6-year-old student TLG started practicing multiplication on an industrial scale. Her current throughput is one 5x5 operation in 20 minutes, and she is improving. She is also learning to troubleshoot her computations and fix mistakes,

Please don't think I am about to teach you multiplication. I am just going to remind you what it takes.

TLG has known multiplication and division since she was 4, but it was not in positional notations. Understanding positional multiplication took her 2 years of my lazy lessons. Not all of them of course, and not even the biggest part of them.

After the initial spatiotemporal training, she learned positional addition in shnumbers. Then she went to numerals working (mostly) with random 10-digit numbers and using the charts of her own making. She learned addition "facts" while adding. Unfortunately, I did not count those 10-digit additions. Looks like she performed less than 30 of them.

According to my homegrown strategy, multiplication teaches addition much better than addition teaches itself. The same is true for long additions teaching one-digit additions.

More than a year ago, TLG had created her first "times table". Sure enough, she filled it out using repeated additions.

TLG has learned exponentiation and developed rudimentary exponential thinking. Not only decimal, and I regularly expose her to non-decimal arithmetic.

I spent couple week's lessons playing with the elements of combinatorics. TLG may even remember factorial.

TLG is not really keen to study math, but she has the right attitude. As soon as she learned to understand my English, I kept telling her: we don't need these patterns, we build them because they are hard to build. She wouldn't run where she is supposed to swim. She only uses the calculator to verify her solutions. She even finds building 10 or 11 digit numbers by hand somewhat exciting.

TLG understood distributivity and remembers it well. In anything remotely similar to an alphabet, she could do transformations like

(A1 + A2)×M = A1×M + A2×M,

or

(A1 + A2)×(B1 + B2) = (A1 + A2)×B1 + (A1 + A2)×B2 
                   = A1×B1 + A2×B1 + A1×B2 + A2×B2

or instantly

(A + B)×(E + F + G) = AE + AF + AG + BE + BF + BG

TLG knows that

NM×NK = NM+K

I'll check on her tomorrow to make sure she still does. She knew how to derive this identity anyway.

TLG knows that positional number means addition of multiplications of powers. She is fairly comfortable with

12345 = 1×104 + 2×103 + 3×102 + 4×101 + 5×100  

Powers of the base are much easier to use than 1s with 0s.

Hence, TLG can take apart two long numbers and multiply them piece-by-piece using distributivity. She can do it in a single line or make a chart matching everything to everything else. Either way, multiplying N-digit number by M-digit number produces N×N partial multiplications, and to "add together" she must rewrite them into a long column.

Approaching the standard algorithm, TLG learned to twist a partial multiplication into a single line. The biggest challenge was to add the carry, and only then to write out the result. Granted, since her early spatiotemporal training, TLG has been very good at such things.

Accumulating or adding together five or more partial additions was easy. We practiced adding and subtracting several long numbers at once.

Trying to teach TLG the standard method of long multiplication, I found several problems. The biggest one was that her handwriting was still too clumsy for small digits. She now starts every multiplication from "setting the stage". This preparatory part on the picture is blue.

On such a stage, TLG has enough room for regular size carries. By the way, we use only 1/2" graph paper. We don't need photocopied forms.

TLG strikes through the carries to get them out of her way when she adds the partial multiplications. I tried several ways to indicate the starting positions and decided to use 0s, but not before TLG has learned to multiply on the Odhner's machine to verify her computations.

I don't care if multiplication is good math or if it's math at all. I accept that TLG may never in her life need to multiply two numbers, and she may forget everything in the next few months. Surely, school will work hard to push her back to the standards of mental developments.

Multiplication for me is a moral and an intellectual challenge. What else in elementary education can match complexity, sophistication and thrill of arithmetic? What can have such an impact on building a character? A child who learned it became a different person.

Geometry—who needs it today? Take it away, and kids will never have a chance to taste a theory.

20160728 PAPER TEACHING

Why educators hate counting?

The picture is from "The Book of Trades". It was created by Jost Amman and published in 1568.

One of the first observations I made trying to understand US elementary school was that the teachers not only sow and nurture what they were told is right - they aggressively prosecute and eradicate what they told is wrong. Weirdly, their major enemy is dactylonomy or finger reckoning.

For this column, I picked a recent article from The Atlantic. The authors try to convince educators to change their attitude. No doubt, school will not listen.

I made several half-hearted attempts to find out exactly when and how finger reckoning became so unwelcome. During antiquity and in the Middle ages it was perfectly fine. It fell out of grace somewhere between 15th and 20th centuries AD.

The obvious reason for expelling finger reckoning from classrooms could be the raise of the Hindu-Arabic numeral system, which we use today. Popularly, it's importation is credited to Leonardo Fibonacci. Apparently, long before him, the wizards of Europe were reading the Latin translations of the works of Muhammad ibn Musa al-Khwarizmi and wondering who could have written them.

Like book printing, calculations in Hindu-Arabic system became popular due to an unsung innovation: Europeans had learned Chinese art of paper making. I heard they started producing paper because they were drowning under rags, and the reason they suddenly faced such an abundance of linen and hemp was the plague.

The next critical change occurred only in 1840s. Industrialists in Germany and Canada learned to make paper out of wood pulp. Their product quickly became cheap enough for disposable news, but schools at that time and and well into the 20th century were using even cheaper substitute: slates.

Their numeracy was about traditional units of money and measurement. Only the numerals were decimal. The technology, however, was modern because calculations were performed in writing. Before writable arithmetic, people had to rely on their fingers and abaci.

The modern school was quite new back then. The king of Prussia Friedrich the Great kick-started the famous Prussian education system in 1763, but only after Napoleonic Wars Europeans took mass education seriously. Napoleon, by the way, promoted decimalisation.

The man behing the Prussian reform was Johann Julius Hecker. He could have been the first to teach teachers. Wikipedia says that the Prussian system "was adopted by all American K-12 public schools and major universities as early as the late 18th century". Did Hecker's seminary teach teachers to root out finger reckoning?

By the middle of the 19th century, elementary education was becoming mandatory. Could this entice educators to crack down on out-of-school knowledge to solidify their emerging monopoly?

Literacy and numeracy, dilapidating cornerstones of US elementary school, are inextricably tied to paper. Even teaching reading using computers, they teach to read from paper. Apparently, at some moment, the teachers decried the shnumber methods and devices like calculi, fingers and abaci to concentrate on selling the only truly scientific technology. Whatever benefits for themselves they saw in it, they cling to paper, pencils and erasers even today, when devices made such a spectacular comeback.

From my American kids, I learned that the teachers secretly told them to use a number line if they can't remember a "math fact". I suspect this device is allowed due to the reasons discussed in this post. School favors stakes: They are more difficult to understand and to use. Devices use sticks. It is unlikely that this school will relearn arithmetic from devices anytime soon. Meanwhile, it's students live in the age of machine computations.

You may think that calculi, fingers, abaci and mechanical calculators were left in the dust. Well, so am I. I use calculi to teach quantity and even algorithms. My and my students' fingers are always with us. The method of positional addition and subtraction in Shnumbers was adapted from decimal abaci, and it does teach numeracy, while elementary school scares students off it. Our Original-Odhner is indispensable for learning multiplication.

Outdated is the whole idea of arithmetic as of elementary math. Outside of schools, this business is called computing.

Meanwhile, paper teachers are struggling with silicon.

20160727 THROUGH PATTERNS TO ALGORITHMS:

Way to go.

Before I forgot, the word algorithm is obviously Arabic, but only recently I learned that it's the name of al-Khwarizmi. The post stamp on the right is Soviet. Al-Khwarizmi was born in Khwarezm, in Uzbekistan, which was a part of USSR.

I started thinking about teaching TLG algorithms thanks to Steven Leinwand. My first reaction was, come on, you guys can't even explain multiplication. Eventually, I decided it was worth trying, and tried.

The material for TLG's first algorithm was bare quantity. I believe, the study of quantity must come before numeracy, and, since numeracy tends to erode the prior knowledge, we must reteach the fundamentals as often as possible.

Quantity is a property of sets of objects. We can't see it (which is truly remarkable). We discover it mentally.

Quantity is something that the sets of countable objects have in common, no matter how different those objects are. I provided some explanation in SHNUMBERS. Now I suspect I will never have a chance to tell everything I want, so please allow me to write just a little bit more.

As counting tokens, calculi or pebbles are second to none. They are small, dirt cheap, low maintenance, rock solid, stone durable, they can be easily differentiated by colors and shapes and even painted. Sadly, people waste plastics buying counting bears, cubes, beads.

As soon as we understood that we can match pebbles to hurricanes one by one and find the number of the pebbles bigger, we are on the way to math. The first step, I believe, is to learn to compare quantities.

I gave TLG two quantities in the paper bags, added one more empty bag, and challenged her to compare without taking all the tokens out. She was upset at first, but quickly understood the she only needed to match two tokens at a time. After she found which quantity was bigger, I started drafting the flowchart. Instantly, TLG learned the language and finished the story.

There is a problem with flowcharts. Let me borrow a piece from CODEPEGS to tell what it is.

Long time ago, when computer programming was in its infancy, we had statements allowing to leap back or forth and continue execution from any point. In 1968 a prominent Mufti named Edsger Wybe Dijkstra published a fatwa, in which he claimed that such statements are evil, and "should be abolished". So abolished they were.

Flowcharts allow to direct the flow of control to any point, so abolished they must be. I did not question the wisdom behind this decision and/or the outcome of it. I simply admit that recently I taught my 6 years 10 months old offspring use three kinds of blocks and lines with arrows to describe algorithms. Moreover, I am going to use flowcharts reteaching her every algorithm she has learned this far (and she is quickly approaching long division).

Three years ago, TLG underwent a course of procedural thinking. Back then, I did not understand what I was teaching her, and how this teaching was linked to my homegrown course of arithmetic. Only recently, I borrowed the unifying concept of spatiotemporal patterns.

Edsger Dijkstra wrote: "My second remark is that our intellectual powers are rather geared to master static relations and that our powers to visualize processes evolving in time are relatively poorly developed."

The "processes evolving in time" are spatiotermporal patterns, and they emerge from the procedures as we play them out. This was what TLG learned by the age of four.

Here is a very basic "checkerboard" pattern, which I temporalized in 6 different ways. I did it using FIREPEGS, which mimics much younger child. A year later the same child would build the same patterns procedurally.

What's next? I already have CODEPEGS. Soon (if not already) TLG may become able to build the patterns through JavaScript.

20160723 CALCULATION, ACCUMULATION, SUMMATION:

Are they addition?

The mystery "add together" operation is known to Wikipedians as summation. You may even see this link on the "Multiplication" page. Well, multiplication is summation of the same number. Why don't they explain it this way?

Summation, according to Wikipedia, to me is no less weird than multiplication. Looks like the writers were struggling with another instance of the "stick and stake" dilemma.

Addition is a binary operation and it involves two stakes. The number of additions is always 1 less than the number of the addends. Wikipedians are trying to avoid instructions like: skip 1, add any two numbers and count 2, add a new number to their sum and count 3... messy, indeed.

There is an obvious solution: begin adding from 0 and count from 1. In

0 + 4 + 4 + 4 

the number of the + signs is equal to the number of the 4s. Why don't they teach this way?

Please start Windows Calculator (I believe, any other calculator will do) and type +4. It becomes 0+4. Now press "=" three times and get 12.

Windows Calculator and the similar devices (including the earliest mechanical calculators and my Original-Odhner) are accumulator machines. You got to clean (or zero) the accumulator before starting using it, so chains of additions start from 0.

In positional notation we omit the leading and the trailing zeros, but we remember they are here. For example,

   1
+100

we understand as

 001
+100

Similarly,

   7
*100

is actually

   7.00
*100.

and we move the point to make 700.

Can we understand

+N

as

0 +N

and

-N

as

0 -N

?

If we could, we could consider addition of three 4s as increment by 4 three times. It might look like

+4 +4 +4 

and sound like add four, add four, add four.

By the way,

0 -N

is negation, or just -N.

Is it strange to add without knowing the augend? To understand how it is possible, consider a simple positional shnumber machine presented in STEREOLEARNING. Click the FORWARD control in the bottom right corner to follow the explanation. A mechanical calculator is simply more complex (and much simpler to use).

By the way, like with any simple calculator, in SHNUMBERS we clean up the board and explicitly add the first quantity to 0. If empty spots and sheets of paper mean 0, we may think we add the first number by writing it. Are they teaching this at schools?

The odometer described by the Roman writer Vitruvius was supposed to count the distances dropping the pebbles in the accumulator bucket. One may imagine dropping several pebbles at a time. To accumulate those pebbles (or calculi), there was absolutely no need to know how many pebbles already were in the bucket. This was true addition.

I wonder why the engineers of Antiquity used the pebbles at all. Why didn't they draw the numbers on the wheel and take the readings? Was it easier to bring the bucket to the masons and order "this many" milestones?

The shnumber devices like abaci and calculators do use the augends. In a sense, they discover them while adding. The reason is: this devices receive the inputs and return results using numerals and/or positional notations.

Teaching unorthodox notations like

+4 +4 +4 

is not a wise policy for a parent, but I have a story to tell. 3 years ago I was not sure how to teach TLG's big sister long multiplication. I didn't see much problems with teaching it, I just wanted to train her in a method endorsed by the educators in her country. It seemed that educators in the US were not endorsing any method and abandoned any hope to teach multiplication at all.

Eventually, I used a periodic discount to buy the video course "Mastering The Fundamentals of Mathematics" by professor James A. Sellers. I did not know US universities teach elementary math. It turned out, they do, and they do know multiplication.

My biggest concern was how to add the partial multiplications. I did not remember what they taught us at school, but I suspected in the US they were doing it differently. The writing media in my school were dip pens (complete with portable inkwells, believe it or not). US schools were relying on pencils and erasers.

I learned a lot from that video course. First of all, professor Seller told me that fifteen times three is fifteen three times. Then, to my astonishment, I realized that he was using pencil and eraser like any elementary school teacher. He was writing the carries in the same place, on top of the numbers. I could not believe they didn't let the students to keep track of their mistakes.

The biggest of his multiplications was 3x3 (456x123). When it came to adding 3 "long" numbers, I was holding my breath. OK, said professor, I taught you guys this stuff in my first lecture about addition.

He only taught addition of 2 long numbers. I had to assume that adding 3 long numbers at once was not in his curriculum at all.

Last Autumn, when I asked TLG to do long addition, she said she wanted to add 3 numbers. I muttered that three numbers are two numbers plus one, and only then I clicked. Adding "together" five or six long numbers, position by position, in accumulator mode: Yes, I did practice this. It was 51 or 52 years ago. So much progress in such a short time.

I picked two 10-digit numbers from TLG's stock and multiplied them by hand. My calculation is in the beginning of this page. I found only one acceptable way to keep track of the carries, and I suspect I would have done better with a dip pen. I simply did not have it. I used a ballpoint pen.

20160722 ADDING TOGETHER:

A mystery ghost operation spotted.

US education is practically impossible to discuss. For years, I was trying to figure out about how schools teach multiplication. This helped me understand how to defend the kids from this teaching, but not to explain why do I think I have to defend them. The past observations worth nothing. Next time there will be another school, another teacher and another curriculum. The damage will be done, and only then I will have the valid reasons for defense.

That's why I am going to quote Wikipedia. To preserve their formatting, I captured the following from my browser's window.

My first question is: how multiplication can be defined as elementary if it is made out of additions? Even though, to put it gently, "may be thought as" is not a definition.

US schoolkids are being trained to add using math facts. The statements or expressions like "adding as many copies of them" immediately invoke a question: adding to what?

The example of multiplying 3 by 4 in the following sentence makes the writers change the words: just "adding" became "adding together". I wonder if they actually teach an operation of "adding together". The reason is clear: they claim they add 3 times, but there are only 2 additions. Addition was not working for them. They tried to conjure up something else.

In my engineering view, those "elementary" operations exist only in teachers' imagination. Under the cover of arithmetic, we actually deal with 1s. To add means exactly to bring two sets of 1s together. Positional addition is only a method to handle big quantities of 1s quickly and easily.

Another remarkable aspect of this "definition" is the use of the notion of copying. Copying would make sense for quantities but not for numbers, unless the writers meant photocopying those numbers. A copy of a number is the same number. In multiplication we add the same numbers, not the copies of them.

I have no way to prove where those "copies" came from, but I suspect that they are creations of teachers' guilty conscience. The educators must understand that their elementary operation of multiplication is performing an impossible trick producing the quantities out of thin air. You have 4 dollars and you need more of them. Multiply by 3, and you have 12. "Copying" may be the teacher's way to explain how this can be possible. You just make the copies of your dollars.

The musing about "not very meaningful" distinction between the order of the multiplier and the multiplicand is superb. No, I have no mercy for the people who teach innocent children this stuff. Fortunately, they don't need my mercy.

20160720 THE SHNUMBER CRUNCHER:

A toy from the past.

Education is famously retarded. Schools always teach the citizens of tomorrow using yesterday's technologies. Not sure about this particular case, but usually they blame their retardations on the kids.

Interestingly, some tools of the past make very desirable and educating toys, if only they were explicitly repurposed. Teaching riding a horse instead of driving a car would not do much good, but how about a pet horse?

Here lies the problem. The tools of the past usually do not fit the economy of toys. I am trying to teach TLG vector graphics. A toy plotter would be great to have. Some are still available, and the used ones on eBay are not very expensive. I doubt if any of them accept SVG (how about Logo?), not to mention the interface, the driver and the supplies.

If anybody would take a risk to design such a toy and put it on the market, it would probably be too expensive. A toy is a one-time thing. Two times, maybe. The lesson will be learned quickly.

Several years ago I paid close to $300 for a well used Odhner Arithmometer on eBay. After some cleaning and oiling, it works acceptably well. Not sure if I would buy it again, but I am glad that once I did it.

The Odhner Atithmometer was a desktop mechanical calculator. Wikipedia has an article on it. Mine is newer than anything pictured there: The model number is 227.

At school, they gave us a brief course of using Arithmometer, and I did not quite get it. I recalled it when I was trying to teach TLG's big brother multiplication. It was hard, and suddenly I wished I had an Odhner.

It turned out, the Odhners were not common it the US and hard to come by. The idea was haunting me though. I finally bought one when I started teaching TLG' big sister. By that time I knew why I wanted it. The Odhner Arithmometer was a shumber machine. Even my counting apps from SHNUMBERS had a similar interface.

As a pet, the Odhner Arithmometer has many nice features. For example, subtracting 35 from 0 it returns 9999999999965. Unfortunately, this little thing weights 12 lbs, so using it requires my constant supervision.

How a device covered in numbers can be a shnumber machine? Remember, my boards are marked with numbers too. Shnumbers embody quantity. Numbers symbolize it. Mechanical calculators like Odhner's don't use "math facts". The quantities are built in the pinwheels.

The Odhner Arithmometer is great for learning because to multiply a number on the cursors (levers) by N, one have to rotate the crank handle exactly N times (I wish every child would do it to understand what "number crunching" means). Isn't it how it should be? It is, but it isn't. I'll take a closer look next time.