### Vn125.1 **Multiplying at Logo; Infinity ** 3/13 & 15/78

3/13 Recently, in response to his studying multiplication at school,

Robby has asked me several times to “test” him on times problems —

i.e. listen while he recites in series a set of products of one digit

with the numbers 1 through 12. I have helped him but also objected that

one never really encounters problems of that sort — one always multiplies

one random digit by another. When asked to set these problems

for him, I offered instead to modify the addition program ADD.WHAT

Miriam formerly used, to pose multiplication problems. Robby said he

would like that.

By changing two expressions, I made ADD.WHAT into TIMES.WHAT —

but that proved too cumbersome a name so it’s now shortened to T. As

Robby executed T (doing problems such as 5 times 7, gleefully triumphant

when T proposed 0 times 6), Miriam, who sat at the next terminal,

asked to use the same program.

The first problem she encountered was 5 x 7. I expected her to

5-count 7 times under finger control. Instead, Miriam asked to borrow

my pen and have some paper, whereon she wrote:

7 + 7 + 7 + 7 + 7 =

Miriam reached the intermediate result 28 (by adding 14 plus 14) but

encountered difficulty with the last 7. Robby told her the result was

35, but she didn’t accept his interference. Eventually she worked out

35 herself (it may have been by counting up, but I am not certain).

The next problem was 3 times 8. Despite my injunction, Robby told

Miriam the result was 24. She did not accept his answer and proposed,

herself, 38 as the result even after stating that two 8’s were 16.

3/15 While I was out of the room, I left my notebook of computation

transcriptions open to page 3 of Arithmetic from Miriam at 6. Upon

returning, I found Miriam reading the book. She then asked my per-

mission, and I said there was no reason not to. At supper this evening,

Miriam asked to play the game “I know a number that’s bigger than yours”

as described on page 3 of Arithmetic. This reconstruction is accurate

in content though not a transcription:

Bob | A thousand. |

Miriam | A thousand plus one (laughing). |

Bob | A million. |

Miriam | A million plus 1. (The laughter builds) |

Bob | You understand, don’t you? A google-plex. |

Miriam | A google-plex plus 1. |

Robby | No. Two google-plexes is bigger. |

Bob | That’s right. |

Miriam | But for any number you can always plus 1 to it. |

Bob | What’s the biggest number of all then? |

Miriam | There isn’t any. You can always plus 1. |

3/22 *Plus ça change plus c’est la meme chose*. Miriam brought a friend,

Laurie Ann, to Logo today. Laurie decided to play with MPOLY (when she

finds a design she likes, Laurie prints the figure, then takes the

drawing home to color in.) Miriam refused to work with Laurie and asked

to use the ‘T’ procedure. The first problem posed to her was 7 times 9.

On the piece of scrap paper preserved here as Addendum 125 – 1, and

working by herself, Miriam “solved” the times problem by encoding it in

hash marks and counting. When the procedure refused to accept her

result of 66, Miriam complained. I remarked that her answer was close

but not perfect. She tried 67. “Colder.” Miriam continued with 65,

64, and finally 63.

Miriam’s next problem was 5 times 7. I went to work with Laurie.

Returning a little later, I saw Miriam keying numbers in sequence.

She had begun at 1 and was trying every number to find the product of

5 times 7. I was amazed and laughed, probably hurting Miriam’s feelings.

She stopped there, at 21.

This next morning, Miriam agrees she would like a TINY.T procedure,

one that would let her pick out the biggest number to be multiplied.

I promised to write such a procedure at our next visit to Logo.

**Relevance**

Most striking in Miriam’s use of T was her clear appreciation that

Robby was using a means of generating correct answers she did not grasp.

She did not immediately recognize that Robby was reciting well-known

results. Instead she cast about for a simple generative procedure which

produces a double-digit result from two single digits. Catenation was

her best speculation.

Six months ago, in __Arithmetic Final__, (Miriam at 6 and 1/2), Miriam

asserted that “infinity” was the biggest number of all (I believe she

heard this piece of impressive knowledge from another first grade

child). After reading the transcript, she came away with a different

point of view (but one not clearly spelled out there) — if one can

always add 1 to a number, can’t one always subtract 1 from a number?

This may be her best road to inventing the negative numbers.

Finally, I find Miriam’s return to the counting of hash marks the

ultimate witness on the centrality of counting in her worlds of

computation. It will be most interesting to see how her knowledge of

multiplication develops.

**Addendum 125-1**

**Multiplication with Hash Mark Counting**