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.