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\usepackage{latexsym}
\usepackage{amsmath}
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\usepackage{amscd}
\usepackage{amsfonts}
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\newcommand{\pedagogy}[1]{\footnote{#1}}
\newcommand{\sirdot}{\hbox{$\odot$}}
\newcommand{\apple}{\hbox{$\Delta$}}
\newcommand{\sirpent}{\pentagon}
\newcommand{\snake}{\hbox{{\bf S}}}

\begin{document}

\title
{
SSEGM
}
\author{
Ethan\ D.\ Bolker \\
Carrie\ Hollingsworth\\
Grace\ Biondi\\ 
Solomon\ Bixby\\
Sam\ Feuer\\
Max\ Schleifer\\
Eva\ Steinitz 
}

\date{ several years ago }

\maketitle

\section*{May 21, 2010 -- naming numbers%
\pedagogy{ I wanted to introduce duodecimal arithmetic. When I've
tried this before with other groups of kids (math club at the Manning,
Sam once) I discovered that one of the major problems was
distinguishing between numbers and their names. Since I'd had occasion
earlier with this group to see how excited they were by choosing names
for mathematical entities, I started with that.
}
}

``For a change, let's all work together, and, for a change, try
to pay attention from time to time rather than rushing excitedly off
to try your own ideas as soon as you get them.''

All (reluctantly): ``OK.''

``We'll start by naming the numbers, and choosing a symbol to
write for each one. For the first few we have the usual names and
symbols:
\begin{center}
0 1 2 3 4 5 6 7 8 9
\end{center}

We don't want to write `10' next, since that uses two of the symbols
over again.''

Sam: ``How about B?''

``Good idea.%
\pedagogy{
I wondered why he started with 
`B' rather than `A', and was tempted to revise his suggestion so that
we could get to standard hexadecimal later on, but decided just to let
it run.
}
Then we could continue this way
\begin{center}
0 1 2 3 4 5 6 7 8 9 B C D E \ldots
\end{center}
But that's boring. Let's make up symbols instead.''

Eva: ``How about a circle with a dot in it for after 9? We can call it
`cirdot'?''

I write \sirdot. Eva salutes Sir Dot. I write ``sirdot'' with an `s'
rather than a `c'.  

Grace: ```Pie' next.'' She sketches a complex picture of a tall
apple pie.

``I think that will take too long to draw each time we need
it. Symbols should be easy to write. How about a piece of pie instead?'' 
I draw \apple. 

Grace: ``Let's call it `apple', not `pie'.''

Max: ``We'll have `sirpent' next.'' He draws \sirpent.

Sam (picking up on ``serpent''): ``Then we can have a snake. It can be
just \snake.''

Next we got a stick figure, ``sirbob,'' and a light bulb, ``sirbulb.''

We could clearly have gone on happily punning new knight number
names. But I wanted to move on to mathematics, so I posed some
arithmetic questions to which they rapidly provided the answers:
$$
3 + 5 = 8
$$
$$
6 + 5 = \apple
$$
$$
8 + 5 = \snake
$$
$$
\sirdot + 2 = \sirpent
$$
Then I asked for 
$$
\sirdot + \sirpent
$$
and it dawned on everyone (particularly Eva) that we would need more
names -- in fact, names forever, one for each number.

I asked how we managed everyday arithmetic with just ten symbols.

``After 9 we write 10.''

``Why do you think we start naming the numbers with two digits after we
reach 9?''

``Because we have just 10 fingers.''

``So if we had 6 fingers on each hand we'd have invented two more
single digit symbols before we moved to two digit numbers. Perhaps

\begin{center}
0 1 2 3 4 5 6 7 8 9 \sirdot \  \apple
\end{center}

Then how would you write the next number?''

All agree on a `1' followed by a `0'.

``And what would you call it?''

Sam blurts out ``ten'' and I admonish him: ``No ordinary names for the
numbers beyond 9, or we'll soon be very confused!''

Grace: ``We can call it `table'.''%
\pedagogy{I thought of suggesting ``douze'', reminiscent of ``dozen'',
but decided not to interrupt whatever they chose to invent.
}

Then we know how to keep counting: ``table 1, table 2 and so
on. What's after ``table 9''?

All: ``table sirdot, table apple, two table.'' I write down
\begin{center}
19, 1\sirdot, 1\apple, 20.
\end{center}

Saying the numbers before writing them guaranteed that they'd say
``two table'' and not ``twenty'' for 20.%
\pedagogy{
I did quietly point out the likely etymology of the English ``twenty''
as  ``two ten'' but no one was paying attention, which was just as
well.
}

They decide they want to name the whole number system as well as
naming the numbers themselves. Someone suggests ``SEGM'' for ``Sam,
Eva, Grace, Max'' and all agree.

\section*{The addition table}

It's time to be systematic and write out SEGM's addition table.
They each start with an empty table of the right size in their math
notebooks and begin to fill it in. 

\vspace{0.2in}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$+$&0&1&2&3&4&5&6&7&8&9&\sirdot&\apple\\
\hline
\hline
$0$&0&1&2&3&4&5&6&7&8&9&\sirdot&\apple\\
$1$&1&2&3&4&5&6&7&8&9&\sirdot&\apple&10\\
$2$&2&3&4&5&6&7&8&9&\sirdot&\apple&10&11\\
\multicolumn{13}{l|}{$\cdots$} \\
\hline
\end{tabular}
\end{center}
\vspace{0.2in}

They know that the columns will be the same as the rows. They quickly
see the constant diagonals that slope down to the left. Then
they busily fill in the numbers without further thought. I suggested
we move on to something more interesting.%
\pedagogy{
I think they would still have to pause for thought to compute $\sirdot
+ \sirdot$, but decide not to spend more time on the full table now.
}
\section*{The multiplication table}

This begins innocently enough. The first two rows are easy. They have
fun with the third row when they see the pattern. Most realize quickly
that counting by twos rather than multiplying is the faster way to get
the next entry. 

\vspace{0.2in}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\times$&0&1&2&3&4&5&6&7&8&9&\sirdot&\apple\\
\hline
\hline
$0$&0&0&0&0&0&0&0&0&0&0&0&0\\
$1$&0&1&2&3&4&5&6&7&8&9&\sirdot&\apple\\
$2$&0&2&4&6&8&\sirdot&10&12&14&16&18&1\sirdot\\
\multicolumn{13}{l|}{$\cdots$} \\
\hline
\end{tabular}
\end{center}
\vspace{0.2in}

Max says ``sirdot is an even number!'' I applaud.
\pedagogy{
Actually, he said apple was an even number, and at the time I didn't
see his error. But the applause is warranted despite the error. What
was important was his realization that \apple\ and \sirdot\ were just
numbers with the usual properties of numbers, independent of what they
were called. He clearly understood ``even'' as a property divorced
from the last digit in decimal representation.
}

The rows for 3 and 4 are harder, but still show patterns:

\vspace{0.2in}
\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\times$&0&1&2&3&4&5&6&7&8&9&\sirdot&\apple\\
\hline
\hline
\multicolumn{13}{l|}{$\cdots$} \\

$3$&0&3&6&9&10&13&16&19&20&23&26&29\\
$4$&0&4&8&10&14&18&20&24&28&30&34&38\\
\multicolumn{13}{l|}{$\cdots$} \\
\hline
\end{tabular}
\end{center}

\vspace{0.2in}
 
They find the row for 5 weird and difficult. No pattern at all. Eva
said, with surprise and pleasure, counting on her fingers (just ten of
them) ``This is hard!'' But the row for 6 is a reward. 

\vspace{0.2in}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|}
%\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\times$&0&1&2&3&4&5&6&7&8&9&\sirdot&\apple\\
\hline
\hline
\multicolumn{13}{l|}{$\cdots$} \\

$5$&0&5&\sirdot&13&18&21&26&2\apple&34&39&42&47\\
$6$&0&6&10&16&20&26&30&36&40&46&50&56\\
\multicolumn{13}{l|}{$\cdots$} \\
\hline
\end{tabular}
\end{center}
\vspace{0.2in}

Looking over Eva's shoulder, I show her that counting by 6's in SEGM
is just like counting by 5's in ordinary English.

I ask them to finish the multiplication table for homework.%
\pedagogy{I wonder if they've done it.}

Here it is with one more row:

\vspace{0.2in}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|}
%\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\times$&0&1&2&3&4&5&6&7&8&9&\sirdot&\apple\\
\hline
\hline
$0$&0&0&0&0&0&0&0&0&0&0&0&0\\
$1$&0&1&2&3&4&5&6&7&8&9&\sirdot&\apple\\
$2$&0&2&4&6&8&\sirdot&10&12&14&16&18&1\sirdot\\
$3$&0&3&6&9&10&13&16&19&20&23&26&29\\
$4$&0&4&8&10&14&18&20&24&28&30&34&38\\
$5$&0&5&\sirdot&13&18&21&26&2\apple&34&39&42&47\\
$6$&0&6&10&16&20&26&30&36&40&46&50&56\\
$7$&0&7&12&19&24&2\apple&36&41&48&53&5\sirdot&65\\
$8$&0&8&14&20&28&34&40&48&54&60&68&74\\
$9$&0&9&16&23&30&39&46&53&60&69&76&83\\
$\sirdot$&0&\sirdot&19&26&34&42&50&5\sirdot&69&86&94&\sirdot 2\\
$\apple$&0&\apple&1\sirdot&29&38&47&56&65&74&83&92&\sirdot 0\\
$10$& 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 & \sirdot 0 &
\apple 0  \\
\hline
\end{tabular}
\end{center}
\vspace{0.2in}

\section*{June 18 -- think in SEGM}

Sam comes early. I give out copies of the first part of this
document. He writes after the last footnote: ``I did it.'' When I ask
where it is he blushes and says he left his math notebook home. So I
add to his text ``but I forgot to bring it.''

When the others arrive it's clear they have done their homework too,
though they haven't brought theirs either.

We play Buzz. Eva and Grace know the game. They immediately begin to
play -- in SEGM: 

``1, 2, buzz, 4, 5, buzz, 7, 8, buzz, \sirdot, \apple, buzz, table 1,
table 2, buzz''

Eva sees that it's easy. She and Grace are alternating the buzzes,
but, more importantly, the last digit tells the whole story.

I ask ``Which numbers are the easy ones in our ordinary language?''

``Just two and five.''

``How about in SEGM?''

All look at the multiplication table I've provided. Two, three, four
and six are easy. They think \sirdot\ and \apple\ will
be hard.

It was late June and warm and that's all we did.

\section*{June 8, 2012 -- SEGM redux}

The four originators have been clamoring for more SEGM for two
years. I've been reluctant, since Sol wasn't in the first group and I
didn't want to leave him out.

But now I've promised fractions and decimals. He can catch up in
advance. It's SSEGM from now on.

Watch this space for a transcript of the recording Carrie made. 
\end{document}
