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.